| L(s) = 1 | + 2-s + 1.20·3-s + 4-s + 2.96·5-s + 1.20·6-s + 2.55·7-s + 8-s − 1.55·9-s + 2.96·10-s + 2.82·11-s + 1.20·12-s + 5.04·13-s + 2.55·14-s + 3.56·15-s + 16-s − 1.93·17-s − 1.55·18-s + 1.22·19-s + 2.96·20-s + 3.07·21-s + 2.82·22-s − 6.25·23-s + 1.20·24-s + 3.79·25-s + 5.04·26-s − 5.47·27-s + 2.55·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.693·3-s + 0.5·4-s + 1.32·5-s + 0.490·6-s + 0.966·7-s + 0.353·8-s − 0.518·9-s + 0.938·10-s + 0.851·11-s + 0.346·12-s + 1.39·13-s + 0.683·14-s + 0.920·15-s + 0.250·16-s − 0.469·17-s − 0.366·18-s + 0.281·19-s + 0.663·20-s + 0.670·21-s + 0.602·22-s − 1.30·23-s + 0.245·24-s + 0.759·25-s + 0.988·26-s − 1.05·27-s + 0.483·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.831006935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.831006935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 - 7.33T + 41T^{2} \) |
| 43 | \( 1 + 6.17T + 43T^{2} \) |
| 47 | \( 1 + 0.107T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 5.98T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.351211430874425657914067835937, −7.938728513985587509472830026089, −6.66950383412958283853856427423, −6.13103874182003110436918634653, −5.57949987678832007699805085193, −4.64724438390058924996024690851, −3.81644654467320204593727088244, −2.96782842825253589959700041247, −1.94936160740429086312099115976, −1.47411033878163865090611939519,
1.47411033878163865090611939519, 1.94936160740429086312099115976, 2.96782842825253589959700041247, 3.81644654467320204593727088244, 4.64724438390058924996024690851, 5.57949987678832007699805085193, 6.13103874182003110436918634653, 6.66950383412958283853856427423, 7.938728513985587509472830026089, 8.351211430874425657914067835937
