L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 4·5-s − 2·6-s − 3·7-s + 9-s + 8·10-s + 4·11-s − 2·12-s − 6·14-s − 4·15-s − 4·16-s + 17-s + 2·18-s − 4·19-s + 8·20-s + 3·21-s + 8·22-s + 6·23-s + 11·25-s − 27-s − 6·28-s − 2·29-s − 8·30-s + 7·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s − 0.816·6-s − 1.13·7-s + 1/3·9-s + 2.52·10-s + 1.20·11-s − 0.577·12-s − 1.60·14-s − 1.03·15-s − 16-s + 0.242·17-s + 0.471·18-s − 0.917·19-s + 1.78·20-s + 0.654·21-s + 1.70·22-s + 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.13·28-s − 0.371·29-s − 1.46·30-s + 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.991931575\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.991931575\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−7.18001928848926202196818434768, −6.62434591070742865786001000296, −6.03350063023248484281217718242, −5.99470992030413824442834128569, −5.01230472509824487666490935934, −4.43938784053246640529869021016, −3.53284332444326021543099477746, −2.80591976345159611591022017208, −2.02637207062549450380972379551, −0.921845718399288912081213009763,
0.921845718399288912081213009763, 2.02637207062549450380972379551, 2.80591976345159611591022017208, 3.53284332444326021543099477746, 4.43938784053246640529869021016, 5.01230472509824487666490935934, 5.99470992030413824442834128569, 6.03350063023248484281217718242, 6.62434591070742865786001000296, 7.18001928848926202196818434768