| L(s) = 1 | + 2.17·3-s − 0.741·5-s + 7-s + 1.74·9-s + 11-s + 6.91·13-s − 1.61·15-s − 7.27·17-s + 4.35·19-s + 2.17·21-s + 3.09·23-s − 4.45·25-s − 2.74·27-s − 3.83·29-s − 10.5·31-s + 2.17·33-s − 0.741·35-s + 2.74·37-s + 15.0·39-s + 7.27·41-s − 6.35·43-s − 1.29·45-s − 3.43·47-s + 49-s − 15.8·51-s + 1.48·53-s − 0.741·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 0.331·5-s + 0.377·7-s + 0.580·9-s + 0.301·11-s + 1.91·13-s − 0.416·15-s − 1.76·17-s + 0.999·19-s + 0.475·21-s + 0.645·23-s − 0.890·25-s − 0.527·27-s − 0.712·29-s − 1.89·31-s + 0.379·33-s − 0.125·35-s + 0.450·37-s + 2.41·39-s + 1.13·41-s − 0.969·43-s − 0.192·45-s − 0.501·47-s + 0.142·49-s − 2.21·51-s + 0.203·53-s − 0.0999·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894613734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894613734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 0.741T + 5T^{2} \) |
| 13 | \( 1 - 6.91T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 3.83T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 - 1.48T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 8.40T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 4.57T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 0.386T + 97T^{2} \) |
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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
−11.34599066892193024627909126840, −11.07680006310362060239204796059, −9.421287975559228696494609430797, −8.846778275545068922937410317921, −8.081731312542757679368832380004, −7.08878816501079040308285094333, −5.78547542369211224423224850811, −4.16545178166705223662181081057, −3.33850054711156013910832543611, −1.81218389160224456671444937759,
1.81218389160224456671444937759, 3.33850054711156013910832543611, 4.16545178166705223662181081057, 5.78547542369211224423224850811, 7.08878816501079040308285094333, 8.081731312542757679368832380004, 8.846778275545068922937410317921, 9.421287975559228696494609430797, 11.07680006310362060239204796059, 11.34599066892193024627909126840
