| L(s) = 1 | − 4-s + 2·9-s + 6·11-s + 16-s + 2·19-s − 12·29-s + 16·31-s − 2·36-s + 6·41-s − 6·44-s + 16·61-s − 64-s − 2·76-s + 20·79-s − 5·81-s − 12·89-s + 12·99-s + 24·101-s + 32·109-s + 12·116-s + 5·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s + 0.458·19-s − 2.22·29-s + 2.87·31-s − 1/3·36-s + 0.937·41-s − 0.904·44-s + 2.04·61-s − 1/8·64-s − 0.229·76-s + 2.25·79-s − 5/9·81-s − 1.27·89-s + 1.20·99-s + 2.38·101-s + 3.06·109-s + 1.11·116-s + 5/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.195293968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.195293968\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 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
−9.135358819577351353868073741156, −8.757989985139321194559377703516, −8.525930594694702119438142975965, −7.946265066587679293000386489286, −7.53224740184146996101872991066, −7.36779537005853799011759317863, −6.63006962575964294137486217517, −6.53292736421301112277410238816, −6.17572318606488773303578170817, −5.45121032642623172074765862558, −5.36574396455507301763712972631, −4.58568863912510922072376849180, −4.19607248178235328252457002854, −4.14812407658660174269129803342, −3.37977632912448211853449947963, −3.20886230013467388552576333118, −2.25156853342336129327138218927, −1.88178797311675741582077054950, −1.06432921178750655888686347073, −0.75439946455104849650220958434,
0.75439946455104849650220958434, 1.06432921178750655888686347073, 1.88178797311675741582077054950, 2.25156853342336129327138218927, 3.20886230013467388552576333118, 3.37977632912448211853449947963, 4.14812407658660174269129803342, 4.19607248178235328252457002854, 4.58568863912510922072376849180, 5.36574396455507301763712972631, 5.45121032642623172074765862558, 6.17572318606488773303578170817, 6.53292736421301112277410238816, 6.63006962575964294137486217517, 7.36779537005853799011759317863, 7.53224740184146996101872991066, 7.946265066587679293000386489286, 8.525930594694702119438142975965, 8.757989985139321194559377703516, 9.135358819577351353868073741156
