| L(s) = 1 | + 2-s + 4-s − 5-s − 2.39·7-s + 8-s − 10-s + 6.13·11-s + 4.73·13-s − 2.39·14-s + 16-s − 4.73·17-s + 4·19-s − 20-s + 6.13·22-s − 23-s − 4·25-s + 4.73·26-s − 2.39·28-s − 2.73·29-s − 2.13·31-s + 32-s − 4.73·34-s + 2.39·35-s + 5·37-s + 4·38-s − 40-s + 5.34·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.905·7-s + 0.353·8-s − 0.316·10-s + 1.84·11-s + 1.31·13-s − 0.640·14-s + 0.250·16-s − 1.14·17-s + 0.917·19-s − 0.223·20-s + 1.30·22-s − 0.208·23-s − 0.800·25-s + 0.929·26-s − 0.452·28-s − 0.508·29-s − 0.383·31-s + 0.176·32-s − 0.812·34-s + 0.404·35-s + 0.821·37-s + 0.648·38-s − 0.158·40-s + 0.834·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.899704098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.899704098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 11 | \( 1 - 6.13T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 + 2.13T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 + 0.394T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 2.26T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 6.73T + 89T^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−8.621412879368921902454547614741, −7.57738586260070331151970101775, −6.84112584487780543648684781492, −6.24176777672970645071615366501, −5.74058582814556040597877385067, −4.43241547448137513513609968353, −3.81624713336711969679040419897, −3.39894260061633423531324797049, −2.09901512793249418772311188169, −0.923357459254104033506809162795,
0.923357459254104033506809162795, 2.09901512793249418772311188169, 3.39894260061633423531324797049, 3.81624713336711969679040419897, 4.43241547448137513513609968353, 5.74058582814556040597877385067, 6.24176777672970645071615366501, 6.84112584487780543648684781492, 7.57738586260070331151970101775, 8.621412879368921902454547614741
