| L(s) = 1 | − 144·11-s + 40·13-s + 432·23-s + 266·25-s − 400·37-s − 1.44e3·47-s + 718·49-s − 288·59-s + 752·61-s + 720·71-s − 1.00e3·73-s + 2.73e3·83-s + 2.09e3·97-s − 4.03e3·107-s − 4.68e3·109-s + 7.69e3·121-s + 127-s + 131-s + 137-s + 139-s − 5.76e3·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + ⋯ |
| L(s) = 1 | − 3.94·11-s + 0.853·13-s + 3.91·23-s + 2.12·25-s − 1.77·37-s − 4.46·47-s + 2.09·49-s − 0.635·59-s + 1.57·61-s + 1.20·71-s − 1.60·73-s + 3.61·83-s + 2.18·97-s − 3.64·107-s − 4.11·109-s + 5.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.36·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.00546·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5427087252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5427087252\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 266 T^{2} + 45051 T^{4} - 266 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 718 T^{2} + 360291 T^{4} - 718 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 72 T + 3931 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 20 T + 606 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12668 T^{2} + 76201926 T^{4} - 12668 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 24460 T^{2} + 242669334 T^{4} - 24460 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 216 T + 32110 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 14540 T^{2} + 285989334 T^{4} - 14540 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 102478 T^{2} + 4392569283 T^{4} - 102478 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 200 T + 76314 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 4252 T^{2} - 5188552410 T^{4} + 4252 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 142420 T^{2} + 11541001398 T^{4} - 142420 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 720 T + 324178 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 17014 T^{2} + 16994482107 T^{4} + 17014 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 144 T + 156634 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 376 T + 427098 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1096180 T^{2} + 479370498006 T^{4} - 1096180 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 360 T + 717010 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 502 T + 370587 T^{2} + 502 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 86492 T^{2} - 48924623994 T^{4} + 86492 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1368 T + 1059307 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1783868 T^{2} + 1521664522278 T^{4} - 1783868 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1046 T + 2036667 T^{2} - 1046 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57255037480573583350985828352, −7.51955410210445518027003040264, −7.08106157481519135457751010114, −6.84360002344678960345227036794, −6.75585627862294998995108857751, −6.42868904491133738703053107390, −6.33652539813233203529360491103, −5.54615845829771609931402528153, −5.50198327776841990408891356921, −5.12781150969048029937182887201, −5.12045923011973385701338087912, −5.02884692051678786947508877006, −4.80650445437460822544923629146, −4.35340980047322977045162547048, −3.72730151869941342984542205363, −3.59528133547501276180471731834, −2.98878983335121089029480863229, −2.94641020434827757739652459143, −2.94509572742665406196006292898, −2.37408506076277971246997320911, −2.14625649951397757742703708113, −1.37746640031638770450345729436, −1.18386756461899495653022573624, −0.69842932872474438857102378127, −0.13014018809712731139556972988,
0.13014018809712731139556972988, 0.69842932872474438857102378127, 1.18386756461899495653022573624, 1.37746640031638770450345729436, 2.14625649951397757742703708113, 2.37408506076277971246997320911, 2.94509572742665406196006292898, 2.94641020434827757739652459143, 2.98878983335121089029480863229, 3.59528133547501276180471731834, 3.72730151869941342984542205363, 4.35340980047322977045162547048, 4.80650445437460822544923629146, 5.02884692051678786947508877006, 5.12045923011973385701338087912, 5.12781150969048029937182887201, 5.50198327776841990408891356921, 5.54615845829771609931402528153, 6.33652539813233203529360491103, 6.42868904491133738703053107390, 6.75585627862294998995108857751, 6.84360002344678960345227036794, 7.08106157481519135457751010114, 7.51955410210445518027003040264, 7.57255037480573583350985828352
