L(s) = 1 | − 2·5-s − 9-s − 8·11-s − 25-s − 12·29-s − 8·31-s − 20·41-s + 2·45-s − 2·49-s + 16·55-s − 8·59-s − 4·61-s − 24·79-s + 81-s + 20·89-s + 8·99-s + 4·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1/3·9-s − 2.41·11-s − 1/5·25-s − 2.22·29-s − 1.43·31-s − 3.12·41-s + 0.298·45-s − 2/7·49-s + 2.15·55-s − 1.04·59-s − 0.512·61-s − 2.70·79-s + 1/9·81-s + 2.11·89-s + 0.804·99-s + 0.398·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.851059057813114294716576139893, −9.404676613215999233946332963402, −8.956154360361886487202524769823, −8.368259070688319206124819723465, −8.203035708950935373983655346043, −7.67839654102252744609709856331, −7.33526250831731147424647497661, −7.17609614322463757056987667011, −6.35418492804859390939354441409, −5.80941734994912013695025701801, −5.29511194129143547094813093334, −5.20457877827392035258724508329, −4.53013155083203388550760594377, −3.91926678170556330330360908835, −3.20330954501237634905713369263, −3.18905211706540382915751011087, −2.16526019093913173537277476252, −1.75010445711386402868872118949, 0, 0,
1.75010445711386402868872118949, 2.16526019093913173537277476252, 3.18905211706540382915751011087, 3.20330954501237634905713369263, 3.91926678170556330330360908835, 4.53013155083203388550760594377, 5.20457877827392035258724508329, 5.29511194129143547094813093334, 5.80941734994912013695025701801, 6.35418492804859390939354441409, 7.17609614322463757056987667011, 7.33526250831731147424647497661, 7.67839654102252744609709856331, 8.203035708950935373983655346043, 8.368259070688319206124819723465, 8.956154360361886487202524769823, 9.404676613215999233946332963402, 9.851059057813114294716576139893