| L(s) = 1 | − 14·5-s − 17·9-s + 32·11-s − 28·13-s − 154·17-s − 224·19-s + 68·23-s + 45·25-s − 236·29-s − 196·31-s + 346·37-s − 420·41-s − 344·43-s + 238·45-s + 84·47-s − 438·53-s − 448·55-s − 56·59-s + 98·61-s + 392·65-s + 336·67-s + 896·71-s + 966·73-s − 52·79-s − 440·81-s + 392·83-s + 2.15e3·85-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 0.629·9-s + 0.877·11-s − 0.597·13-s − 2.19·17-s − 2.70·19-s + 0.616·23-s + 9/25·25-s − 1.51·29-s − 1.13·31-s + 1.53·37-s − 1.59·41-s − 1.21·43-s + 0.788·45-s + 0.260·47-s − 1.13·53-s − 1.09·55-s − 0.123·59-s + 0.205·61-s + 0.748·65-s + 0.612·67-s + 1.49·71-s + 1.54·73-s − 0.0740·79-s − 0.603·81-s + 0.518·83-s + 2.75·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 17 T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 14 T + 151 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 32 T + 1105 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 28 T + 3998 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 154 T + 15163 T^{2} + 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 224 T + 25929 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 68 T + 23677 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 33694 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 196 T + 67373 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 346 T + 65967 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 420 T + 176614 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 84 T + 201085 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 438 T + 280447 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 56 T + 360889 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 98 T + 253751 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 336 T + 627937 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 896 T + 800494 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 966 T + 187 p^{2} T^{2} - 966 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 52 T + 332261 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 392 T + 895462 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 294 T + 1298347 T^{2} - 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 420 T + 1655734 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−11.50305745965956194702743090768, −11.45695815054120075962035368591, −10.80918552284762590320421967192, −10.76824062622858711036973019571, −9.628636092177406919684394689622, −9.235370152542816825252926220637, −8.738130643281818182171738298682, −8.278551643650360686471859806454, −7.88591283822694986533319180516, −7.01653833286608017197070434460, −6.63069760378476486248722576046, −6.29258833211850308091757562822, −5.25116919918848854530789952459, −4.61844003986363486281745898657, −4.00761387758229104088140634818, −3.67914194475695210836413398750, −2.48274414125859817968103859325, −1.87397472546517377911628955261, 0, 0,
1.87397472546517377911628955261, 2.48274414125859817968103859325, 3.67914194475695210836413398750, 4.00761387758229104088140634818, 4.61844003986363486281745898657, 5.25116919918848854530789952459, 6.29258833211850308091757562822, 6.63069760378476486248722576046, 7.01653833286608017197070434460, 7.88591283822694986533319180516, 8.278551643650360686471859806454, 8.738130643281818182171738298682, 9.235370152542816825252926220637, 9.628636092177406919684394689622, 10.76824062622858711036973019571, 10.80918552284762590320421967192, 11.45695815054120075962035368591, 11.50305745965956194702743090768
