| L(s) = 1 | − 2-s − 2.80·3-s + 4-s + 1.91·5-s + 2.80·6-s − 1.55·7-s − 8-s + 4.85·9-s − 1.91·10-s + 5.66·11-s − 2.80·12-s − 3.83·13-s + 1.55·14-s − 5.36·15-s + 16-s + 0.620·17-s − 4.85·18-s + 6.54·19-s + 1.91·20-s + 4.36·21-s − 5.66·22-s − 2.62·23-s + 2.80·24-s − 1.33·25-s + 3.83·26-s − 5.19·27-s − 1.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.855·5-s + 1.14·6-s − 0.588·7-s − 0.353·8-s + 1.61·9-s − 0.605·10-s + 1.70·11-s − 0.808·12-s − 1.06·13-s + 0.416·14-s − 1.38·15-s + 0.250·16-s + 0.150·17-s − 1.14·18-s + 1.50·19-s + 0.427·20-s + 0.951·21-s − 1.20·22-s − 0.546·23-s + 0.571·24-s − 0.267·25-s + 0.752·26-s − 0.998·27-s − 0.294·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8970067344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8970067344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 - 0.620T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 6.65T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 + 1.03T + 71T^{2} \) |
| 73 | \( 1 + 7.63T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.95T + 83T^{2} \) |
| 89 | \( 1 + 6.66T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.613391433393251117157941781505, −7.35491040488792836592904826295, −6.93241503673178315872594822623, −6.12983516583732254928853912732, −5.82568946144946426328729560498, −4.91904920672737766084764903843, −3.98716236597988889459916275002, −2.73301099101065374573461839487, −1.50346528822246146693193122260, −0.70838928194580813516674092732,
0.70838928194580813516674092732, 1.50346528822246146693193122260, 2.73301099101065374573461839487, 3.98716236597988889459916275002, 4.91904920672737766084764903843, 5.82568946144946426328729560498, 6.12983516583732254928853912732, 6.93241503673178315872594822623, 7.35491040488792836592904826295, 8.613391433393251117157941781505
