| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s + 11-s − 4.74·13-s − 2·15-s + 4.74·17-s − 4.74·19-s + 2·21-s − 4.74·23-s + 25-s + 4·27-s − 2.74·29-s + 6.74·31-s − 2·33-s − 35-s − 10.7·37-s + 9.48·39-s − 4·41-s − 4·43-s + 45-s + 6.74·47-s + 49-s − 9.48·51-s − 1.25·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s − 1.31·13-s − 0.516·15-s + 1.15·17-s − 1.08·19-s + 0.436·21-s − 0.989·23-s + 0.200·25-s + 0.769·27-s − 0.509·29-s + 1.21·31-s − 0.348·33-s − 0.169·35-s − 1.76·37-s + 1.51·39-s − 0.624·41-s − 0.609·43-s + 0.149·45-s + 0.983·47-s + 0.142·49-s − 1.32·51-s − 0.172·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7888905916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7888905916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 1.25T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 0.744T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−8.053310770536038546101415107672, −7.04784912433877600333725961980, −6.62512623226233436129155415369, −5.83767784859191261404034025144, −5.35014471702454459723980171970, −4.65061072071387525111652100457, −3.72374592446915358912035296462, −2.69718163437861589422131314482, −1.76002392868902713310418394936, −0.48171704722400981211966423514,
0.48171704722400981211966423514, 1.76002392868902713310418394936, 2.69718163437861589422131314482, 3.72374592446915358912035296462, 4.65061072071387525111652100457, 5.35014471702454459723980171970, 5.83767784859191261404034025144, 6.62512623226233436129155415369, 7.04784912433877600333725961980, 8.053310770536038546101415107672
