| L(s) = 1 | + 12·5-s − 40·13-s + 16·17-s − 142·25-s − 92·29-s + 328·37-s + 624·41-s − 238·49-s + 532·53-s + 264·61-s − 480·65-s + 492·73-s + 192·85-s + 2.78e3·89-s − 604·97-s + 2.93e3·101-s + 3.12e3·109-s − 3.10e3·113-s − 870·121-s − 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.07·5-s − 0.853·13-s + 0.228·17-s − 1.13·25-s − 0.589·29-s + 1.45·37-s + 2.37·41-s − 0.693·49-s + 1.37·53-s + 0.554·61-s − 0.915·65-s + 0.788·73-s + 0.245·85-s + 3.31·89-s − 0.632·97-s + 2.88·101-s + 2.74·109-s − 2.58·113-s − 0.653·121-s − 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.051788571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.051788571\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 34 p T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 870 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4338 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59134 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 164 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 312 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20186 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 178974 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 346246 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 343478 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 257070 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 931870 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 195606 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1392 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{2} \) |
| 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
−9.068320718430586707576346592331, −8.510203494958569075473569957636, −7.82914768500903149922454862135, −7.76071441139252348984212924569, −7.46117180224875122397741590603, −6.89088889203380375911102492517, −6.39493412954873162441148636136, −6.08390815882679726651876947523, −5.69092748653825105772375589094, −5.50236092771098475603889010913, −4.80189099312086770366939471157, −4.60339266337629697763503323342, −3.86413657686958596916412871263, −3.72186251088187267584349983865, −2.84899815356566449971456843329, −2.55299902155961116719969862621, −1.94524802376396156095771121162, −1.82057798586685079973966681706, −0.70976619479302057083419087927, −0.62323311141488764755729211682,
0.62323311141488764755729211682, 0.70976619479302057083419087927, 1.82057798586685079973966681706, 1.94524802376396156095771121162, 2.55299902155961116719969862621, 2.84899815356566449971456843329, 3.72186251088187267584349983865, 3.86413657686958596916412871263, 4.60339266337629697763503323342, 4.80189099312086770366939471157, 5.50236092771098475603889010913, 5.69092748653825105772375589094, 6.08390815882679726651876947523, 6.39493412954873162441148636136, 6.89088889203380375911102492517, 7.46117180224875122397741590603, 7.76071441139252348984212924569, 7.82914768500903149922454862135, 8.510203494958569075473569957636, 9.068320718430586707576346592331
