| L(s) = 1 | + 5-s + 7-s + 5·11-s − 13-s + 7·17-s + 6·19-s + 3·23-s + 25-s − 2·29-s − 2·31-s + 35-s + 7·37-s − 9·41-s + 8·43-s + 10·47-s − 6·49-s − 5·53-s + 5·55-s + 5·61-s − 65-s + 4·67-s + 9·71-s − 6·73-s + 5·77-s + 3·79-s − 4·83-s + 7·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.50·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s + 0.625·23-s + 1/5·25-s − 0.371·29-s − 0.359·31-s + 0.169·35-s + 1.15·37-s − 1.40·41-s + 1.21·43-s + 1.45·47-s − 6/7·49-s − 0.686·53-s + 0.674·55-s + 0.640·61-s − 0.124·65-s + 0.488·67-s + 1.06·71-s − 0.702·73-s + 0.569·77-s + 0.337·79-s − 0.439·83-s + 0.759·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.263586523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.263586523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−7.61878349053172615196492189101, −7.09314952968165761155616417327, −6.32941819999593895475647798364, −5.55606658713101802692726181024, −5.13894229843152297051037082504, −4.12653514758172088178033802606, −3.46723566963146546148453650052, −2.67121302362689198861789022483, −1.48310015906235085159461688001, −1.00505756298064492869582242688,
1.00505756298064492869582242688, 1.48310015906235085159461688001, 2.67121302362689198861789022483, 3.46723566963146546148453650052, 4.12653514758172088178033802606, 5.13894229843152297051037082504, 5.55606658713101802692726181024, 6.32941819999593895475647798364, 7.09314952968165761155616417327, 7.61878349053172615196492189101
