| L(s) = 1 | + 2.67·3-s − 5-s + 4.67·7-s + 4.14·9-s + 0.672·11-s − 1.14·13-s − 2.67·15-s + 3.52·17-s − 5.52·19-s + 12.4·21-s + 3.81·23-s + 25-s + 3.05·27-s − 29-s + 1.52·31-s + 1.79·33-s − 4.67·35-s + 7.16·37-s − 3.05·39-s + 2.85·41-s − 8.96·43-s − 4.14·45-s + 6.67·47-s + 14.8·49-s + 9.43·51-s + 10.4·53-s − 0.672·55-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 0.447·5-s + 1.76·7-s + 1.38·9-s + 0.202·11-s − 0.317·13-s − 0.690·15-s + 0.855·17-s − 1.26·19-s + 2.72·21-s + 0.795·23-s + 0.200·25-s + 0.588·27-s − 0.185·29-s + 0.274·31-s + 0.313·33-s − 0.789·35-s + 1.17·37-s − 0.489·39-s + 0.446·41-s − 1.36·43-s − 0.617·45-s + 0.973·47-s + 2.11·49-s + 1.32·51-s + 1.44·53-s − 0.0907·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.619925626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.619925626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 - 0.672T + 11T^{2} \) |
| 13 | \( 1 + 1.14T + 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
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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
−8.687680854799441387316241402049, −8.330445210918978505813524479566, −7.65796157152496869288335528901, −7.14482162110104652852548011775, −5.78514484794685701332615633680, −4.65491590974454258454652097099, −4.19656326550927076822799169220, −3.11792587591189350917308135677, −2.22720248327096443703740557520, −1.31277044420626135305584383448,
1.31277044420626135305584383448, 2.22720248327096443703740557520, 3.11792587591189350917308135677, 4.19656326550927076822799169220, 4.65491590974454258454652097099, 5.78514484794685701332615633680, 7.14482162110104652852548011775, 7.65796157152496869288335528901, 8.330445210918978505813524479566, 8.687680854799441387316241402049
