L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s + 2·13-s + 15-s − 16-s + 2·17-s − 18-s − 4·19-s + 20-s − 4·22-s − 3·24-s + 25-s − 2·26-s − 27-s + 2·29-s − 30-s − 5·32-s − 4·33-s − 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.883·32-s − 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4656717163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4656717163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 47 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.96988121065474, −14.56049920496281, −13.93829422224649, −13.42617754646617, −12.90386588182312, −12.13087947031564, −11.93232508554282, −11.22957095901129, −10.58963813868746, −10.32762458061877, −9.588327639083278, −9.079182332929625, −8.525631496212021, −8.159608564829133, −7.410659396144726, −6.774486557591058, −6.358591877837365, −5.589314827332597, −4.802636589904197, −4.425642990265391, −3.677157703038847, −3.184899025362603, −1.631770905444037, −1.466212756471995, −0.3219951926837716,
0.3219951926837716, 1.466212756471995, 1.631770905444037, 3.184899025362603, 3.677157703038847, 4.425642990265391, 4.802636589904197, 5.589314827332597, 6.358591877837365, 6.774486557591058, 7.410659396144726, 8.159608564829133, 8.525631496212021, 9.079182332929625, 9.588327639083278, 10.32762458061877, 10.58963813868746, 11.22957095901129, 11.93232508554282, 12.13087947031564, 12.90386588182312, 13.42617754646617, 13.93829422224649, 14.56049920496281, 14.96988121065474