| L(s) = 1 | − 10·5-s − 2·9-s + 68·13-s + 228·17-s + 75·25-s − 52·29-s − 300·37-s + 684·41-s + 20·45-s − 634·49-s − 524·53-s − 524·61-s − 680·65-s + 1.36e3·73-s − 725·81-s − 2.28e3·85-s − 1.26e3·89-s − 1.93e3·97-s + 3.27e3·101-s − 684·109-s + 4.21e3·113-s − 136·117-s − 790·121-s − 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.0740·9-s + 1.45·13-s + 3.25·17-s + 3/5·25-s − 0.332·29-s − 1.33·37-s + 2.60·41-s + 0.0662·45-s − 1.84·49-s − 1.35·53-s − 1.09·61-s − 1.29·65-s + 2.18·73-s − 0.994·81-s − 2.90·85-s − 1.50·89-s − 2.02·97-s + 3.22·101-s − 0.601·109-s + 3.50·113-s − 0.107·117-s − 0.593·121-s − 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.146312749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.146312749\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 634 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 790 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19398 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 49390 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 47374 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 133526 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 170310 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 353954 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 392610 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 682 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 945310 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1120642 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 966 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.49003558253112920914958284721, −12.38487837713366474078240689695, −11.59891154421004489653131496834, −11.21755171224890145406112500375, −10.79268184632276725132185486410, −10.18308817016433810082649079818, −9.609298784959027436598883974849, −9.172901688584562362713320968989, −8.257862174868890162050513331404, −8.097754676814228269932349434855, −7.56961178871412032587388056431, −6.99163985445905236322984814480, −6.03436319431682710364545563810, −5.77771322950821232215338289878, −4.98884383692948695652089246103, −4.15829131094677255206442025173, −3.28101673064146382899431725068, −3.27458454718604352225616135221, −1.56007874739793814512359953671, −0.76628277641064768699482609404,
0.76628277641064768699482609404, 1.56007874739793814512359953671, 3.27458454718604352225616135221, 3.28101673064146382899431725068, 4.15829131094677255206442025173, 4.98884383692948695652089246103, 5.77771322950821232215338289878, 6.03436319431682710364545563810, 6.99163985445905236322984814480, 7.56961178871412032587388056431, 8.097754676814228269932349434855, 8.257862174868890162050513331404, 9.172901688584562362713320968989, 9.609298784959027436598883974849, 10.18308817016433810082649079818, 10.79268184632276725132185486410, 11.21755171224890145406112500375, 11.59891154421004489653131496834, 12.38487837713366474078240689695, 12.49003558253112920914958284721
