| L(s) = 1 | + 2-s + 1.93·3-s + 4-s + 1.16·5-s + 1.93·6-s + 8-s + 0.730·9-s + 1.16·10-s + 1.40·11-s + 1.93·12-s + 4.56·13-s + 2.25·15-s + 16-s − 4.59·17-s + 0.730·18-s + 1.15·19-s + 1.16·20-s + 1.40·22-s + 6.00·23-s + 1.93·24-s − 3.63·25-s + 4.56·26-s − 4.38·27-s + 4.33·29-s + 2.25·30-s + 7.01·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.11·3-s + 0.5·4-s + 0.521·5-s + 0.788·6-s + 0.353·8-s + 0.243·9-s + 0.368·10-s + 0.424·11-s + 0.557·12-s + 1.26·13-s + 0.581·15-s + 0.250·16-s − 1.11·17-s + 0.172·18-s + 0.265·19-s + 0.260·20-s + 0.300·22-s + 1.25·23-s + 0.394·24-s − 0.727·25-s + 0.895·26-s − 0.843·27-s + 0.804·29-s + 0.411·30-s + 1.25·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.371837036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.371837036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 1.16T + 5T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 4.33T + 29T^{2} \) |
| 31 | \( 1 - 7.01T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 0.473T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 7.11T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 - 3.31T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−8.509464893770501040035536327570, −7.83653727374814559676261470991, −6.72875196493869248432154518164, −6.37075267628119533308964661848, −5.43403570068549774304942020947, −4.52876125621302342831924528035, −3.72831263083728642040709692620, −3.00686443897721175969339370601, −2.24749958297416509134274181232, −1.26020124105450549697162550433,
1.26020124105450549697162550433, 2.24749958297416509134274181232, 3.00686443897721175969339370601, 3.72831263083728642040709692620, 4.52876125621302342831924528035, 5.43403570068549774304942020947, 6.37075267628119533308964661848, 6.72875196493869248432154518164, 7.83653727374814559676261470991, 8.509464893770501040035536327570
