| L(s) = 1 | + 2.48·2-s − 3-s + 4.19·4-s − 2.36·5-s − 2.48·6-s + 5.45·8-s + 9-s − 5.88·10-s + 4.46·11-s − 4.19·12-s + 5.01·13-s + 2.36·15-s + 5.18·16-s − 4.88·17-s + 2.48·18-s + 3.64·19-s − 9.91·20-s + 11.1·22-s − 2.22·23-s − 5.45·24-s + 0.599·25-s + 12.4·26-s − 27-s + 2.88·29-s + 5.88·30-s − 1.18·31-s + 1.98·32-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.09·4-s − 1.05·5-s − 1.01·6-s + 1.92·8-s + 0.333·9-s − 1.86·10-s + 1.34·11-s − 1.20·12-s + 1.39·13-s + 0.611·15-s + 1.29·16-s − 1.18·17-s + 0.586·18-s + 0.836·19-s − 2.21·20-s + 2.36·22-s − 0.462·23-s − 1.11·24-s + 0.119·25-s + 2.44·26-s − 0.192·27-s + 0.535·29-s + 1.07·30-s − 0.212·31-s + 0.351·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.742576614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.742576614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + 0.748T + 37T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 0.0922T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 7.87T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−7.69942196306514256814277874753, −7.07464025358121963993913626205, −6.30944156689323903044334628908, −6.03550259358055130532047727917, −5.05848274789539829116537935913, −4.23652994559839776249167117199, −3.93209264130612321115673514904, −3.30131491293940674303360225095, −2.08202714801418945009548731635, −0.936592332778388759992096938172,
0.936592332778388759992096938172, 2.08202714801418945009548731635, 3.30131491293940674303360225095, 3.93209264130612321115673514904, 4.23652994559839776249167117199, 5.05848274789539829116537935913, 6.03550259358055130532047727917, 6.30944156689323903044334628908, 7.07464025358121963993913626205, 7.69942196306514256814277874753
