| L(s) = 1 | + 2·3-s − 2·4-s − 3·9-s + 6·11-s − 4·12-s + 4·16-s + 12·17-s + 4·19-s − 10·25-s − 14·27-s + 12·33-s + 6·36-s − 18·41-s + 16·43-s − 12·44-s + 8·48-s − 13·49-s + 24·51-s + 8·57-s + 24·59-s − 8·64-s − 8·67-s − 24·68-s + 22·73-s − 20·75-s − 8·76-s − 4·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 9-s + 1.80·11-s − 1.15·12-s + 16-s + 2.91·17-s + 0.917·19-s − 2·25-s − 2.69·27-s + 2.08·33-s + 36-s − 2.81·41-s + 2.43·43-s − 1.80·44-s + 1.15·48-s − 1.85·49-s + 3.36·51-s + 1.05·57-s + 3.12·59-s − 64-s − 0.977·67-s − 2.91·68-s + 2.57·73-s − 2.30·75-s − 0.917·76-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.814042266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.814042266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.671747243929389653949444047888, −9.032345851694468357832029856913, −8.872565863901409987167311116002, −8.123034833885839175467777085057, −7.82035864868172488569600477822, −7.59911177067371416247669368567, −6.52062745028658305026776170860, −5.94767259032431241696446624088, −5.44973416215471530225317007593, −5.05990556060455221641782081661, −3.82056801715987130500814062168, −3.50910294340479626549928321873, −3.39545840765879304672503047561, −2.15016244431519898508252799169, −1.05845859534398233358847781067,
1.05845859534398233358847781067, 2.15016244431519898508252799169, 3.39545840765879304672503047561, 3.50910294340479626549928321873, 3.82056801715987130500814062168, 5.05990556060455221641782081661, 5.44973416215471530225317007593, 5.94767259032431241696446624088, 6.52062745028658305026776170860, 7.59911177067371416247669368567, 7.82035864868172488569600477822, 8.123034833885839175467777085057, 8.872565863901409987167311116002, 9.032345851694468357832029856913, 9.671747243929389653949444047888
