| L(s) = 1 | + 3-s − 3.15·5-s + 4.69·7-s + 9-s − 0.137·11-s − 3.15·15-s − 5.60·17-s + 4.98·19-s + 4.69·21-s + 6.09·23-s + 4.97·25-s + 27-s − 0.850·29-s − 6.23·31-s − 0.137·33-s − 14.8·35-s + 11.7·37-s − 4.27·41-s + 2.09·43-s − 3.15·45-s − 4.98·47-s + 15.0·49-s − 5.60·51-s − 1.82·53-s + 0.432·55-s + 4.98·57-s + 5.89·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.41·5-s + 1.77·7-s + 0.333·9-s − 0.0413·11-s − 0.815·15-s − 1.35·17-s + 1.14·19-s + 1.02·21-s + 1.27·23-s + 0.995·25-s + 0.192·27-s − 0.157·29-s − 1.11·31-s − 0.0238·33-s − 2.50·35-s + 1.92·37-s − 0.667·41-s + 0.319·43-s − 0.470·45-s − 0.727·47-s + 2.14·49-s − 0.784·51-s − 0.251·53-s + 0.0583·55-s + 0.660·57-s + 0.768·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.449197249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.449197249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 0.137T + 11T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 0.850T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 - 0.0978T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.85T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−7.86698458242193521328449356089, −7.36068119899635228621345391635, −6.79422148340942119837430215179, −5.49899560743922865113482615454, −4.80238638213805621284302924001, −4.32634901193769798644573483481, −3.60583307969797927715169294998, −2.70325707116672335674925087520, −1.76663806319342905446693793824, −0.77821818188097693257205406722,
0.77821818188097693257205406722, 1.76663806319342905446693793824, 2.70325707116672335674925087520, 3.60583307969797927715169294998, 4.32634901193769798644573483481, 4.80238638213805621284302924001, 5.49899560743922865113482615454, 6.79422148340942119837430215179, 7.36068119899635228621345391635, 7.86698458242193521328449356089
