| L(s) = 1 | − 6·3-s + 27·9-s + 40·11-s − 68·17-s + 104·19-s − 242·25-s − 108·27-s − 240·33-s − 52·41-s + 504·43-s − 486·49-s + 408·51-s − 624·57-s + 728·59-s + 1.25e3·67-s + 676·73-s + 1.45e3·75-s + 405·81-s + 2.07e3·83-s + 468·89-s − 356·97-s + 1.08e3·99-s + 2.80e3·107-s + 2.75e3·113-s − 1.46e3·121-s + 312·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.09·11-s − 0.970·17-s + 1.25·19-s − 1.93·25-s − 0.769·27-s − 1.26·33-s − 0.198·41-s + 1.78·43-s − 1.41·49-s + 1.12·51-s − 1.45·57-s + 1.60·59-s + 2.29·67-s + 1.08·73-s + 2.23·75-s + 5/9·81-s + 2.74·83-s + 0.557·89-s − 0.372·97-s + 1.09·99-s + 2.53·107-s + 2.29·113-s − 1.09·121-s + 0.228·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.127360523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127360523\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 242 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 486 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2826 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 20462 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8450 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47414 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 27578 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 252 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88574 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 166894 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 364 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86838 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 628 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 604430 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 338 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 363350 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1036 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
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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
−10.00750755086429098712667813092, −9.705020096872218086232039158085, −9.464874960759335128461078658517, −9.065461452630530238400544337328, −8.275462548805946680030689866537, −8.085711879257067165617736158214, −7.23860818730725086101304905575, −7.22886119921474936393841496614, −6.40286643664248732721942221439, −6.36270287274552723992973483062, −5.57694256957080654905616839450, −5.48482207200591377096013439877, −4.59388713153369486940798887609, −4.45071942752871467564195329991, −3.53304058456337200427137459094, −3.52972691140513077770072419225, −2.16357949390786129934228323707, −1.95499855770089715688544576577, −0.891240065564593286217391639990, −0.56654290941226842393521972251,
0.56654290941226842393521972251, 0.891240065564593286217391639990, 1.95499855770089715688544576577, 2.16357949390786129934228323707, 3.52972691140513077770072419225, 3.53304058456337200427137459094, 4.45071942752871467564195329991, 4.59388713153369486940798887609, 5.48482207200591377096013439877, 5.57694256957080654905616839450, 6.36270287274552723992973483062, 6.40286643664248732721942221439, 7.22886119921474936393841496614, 7.23860818730725086101304905575, 8.085711879257067165617736158214, 8.275462548805946680030689866537, 9.065461452630530238400544337328, 9.464874960759335128461078658517, 9.705020096872218086232039158085, 10.00750755086429098712667813092
