| L(s) = 1 | + 7·4-s − 20·5-s − 68·11-s − 15·16-s + 44·19-s − 140·20-s + 275·25-s + 28·29-s − 584·31-s + 204·41-s − 476·44-s + 670·49-s + 1.36e3·55-s − 380·59-s − 508·61-s − 553·64-s − 2.19e3·71-s + 308·76-s + 2.70e3·79-s + 300·80-s − 1.86e3·89-s − 880·95-s + 1.92e3·100-s + 2.01e3·101-s + 1.56e3·109-s + 196·116-s + 806·121-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 1.78·5-s − 1.86·11-s − 0.234·16-s + 0.531·19-s − 1.56·20-s + 11/5·25-s + 0.179·29-s − 3.38·31-s + 0.777·41-s − 1.63·44-s + 1.95·49-s + 3.33·55-s − 0.838·59-s − 1.06·61-s − 1.08·64-s − 3.66·71-s + 0.464·76-s + 3.85·79-s + 0.419·80-s − 2.21·89-s − 0.950·95-s + 1.92·100-s + 1.98·101-s + 1.37·109-s + 0.156·116-s + 0.605·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 174 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1834 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 292 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 24010 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 p^{2} T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6942 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 241110 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 190 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 17830 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1096 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 603310 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1352 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1119238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 930 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1812350 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43262499259107525721455142531, −9.604821998508907056742435756552, −9.047191071055435651713385396361, −8.776256165315784526263178350174, −8.170869945517241440755920828412, −7.48426855409844866782667671811, −7.46759186660069998835869220501, −7.37394607879843762171099996406, −6.62418682624401725859506810273, −5.73686866137063091592899244759, −5.65556480670232496997259235769, −4.74395354163446120493110208123, −4.55905102633063105691736367175, −3.57406224664580857642046551683, −3.44855022033988405167435359748, −2.62192500420184878307665695938, −2.21563860996339263325022802177, −1.20090138175627443923869725718, 0, 0,
1.20090138175627443923869725718, 2.21563860996339263325022802177, 2.62192500420184878307665695938, 3.44855022033988405167435359748, 3.57406224664580857642046551683, 4.55905102633063105691736367175, 4.74395354163446120493110208123, 5.65556480670232496997259235769, 5.73686866137063091592899244759, 6.62418682624401725859506810273, 7.37394607879843762171099996406, 7.46759186660069998835869220501, 7.48426855409844866782667671811, 8.170869945517241440755920828412, 8.776256165315784526263178350174, 9.047191071055435651713385396361, 9.604821998508907056742435756552, 10.43262499259107525721455142531
