| L(s) = 1 | + 1.36·2-s − 0.755·3-s − 0.141·4-s − 5-s − 1.03·6-s + 1.53·7-s − 2.91·8-s − 2.42·9-s − 1.36·10-s + 11-s + 0.106·12-s + 4.94·13-s + 2.09·14-s + 0.755·15-s − 3.69·16-s + 0.774·17-s − 3.31·18-s − 3.33·19-s + 0.141·20-s − 1.16·21-s + 1.36·22-s − 5.98·23-s + 2.20·24-s + 25-s + 6.74·26-s + 4.10·27-s − 0.217·28-s + ⋯ |
| L(s) = 1 | + 0.964·2-s − 0.436·3-s − 0.0705·4-s − 0.447·5-s − 0.420·6-s + 0.581·7-s − 1.03·8-s − 0.809·9-s − 0.431·10-s + 0.301·11-s + 0.0307·12-s + 1.37·13-s + 0.560·14-s + 0.195·15-s − 0.924·16-s + 0.187·17-s − 0.780·18-s − 0.764·19-s + 0.0315·20-s − 0.253·21-s + 0.290·22-s − 1.24·23-s + 0.450·24-s + 0.200·25-s + 1.32·26-s + 0.789·27-s − 0.0410·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853526812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853526812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 + 0.755T + 3T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 0.774T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 5.98T + 23T^{2} \) |
| 29 | \( 1 + 0.646T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 + 4.13T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 0.733T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 4.51T + 83T^{2} \) |
| 89 | \( 1 + 0.270T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.386181362698161171484378693008, −7.86984367414391295144074693464, −6.56170543789753695962782088933, −6.07392739669733139239890255056, −5.47966903172642751185820826840, −4.60996841688776130404000360698, −3.96050076653770641863750320225, −3.31524985954356059437866774620, −2.15116058689516911836215732258, −0.68235305533093118752762571772,
0.68235305533093118752762571772, 2.15116058689516911836215732258, 3.31524985954356059437866774620, 3.96050076653770641863750320225, 4.60996841688776130404000360698, 5.47966903172642751185820826840, 6.07392739669733139239890255056, 6.56170543789753695962782088933, 7.86984367414391295144074693464, 8.386181362698161171484378693008
