| L(s) = 1 | − 2-s − 3-s + 4-s + 1.78·5-s + 6-s − 7-s − 8-s + 9-s − 1.78·10-s + 3.17·11-s − 12-s + 14-s − 1.78·15-s + 16-s + 5.56·17-s − 18-s − 6.18·19-s + 1.78·20-s + 21-s − 3.17·22-s + 6.13·23-s + 24-s − 1.80·25-s − 27-s − 28-s + 2.07·29-s + 1.78·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.799·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.565·10-s + 0.957·11-s − 0.288·12-s + 0.267·14-s − 0.461·15-s + 0.250·16-s + 1.34·17-s − 0.235·18-s − 1.41·19-s + 0.399·20-s + 0.218·21-s − 0.676·22-s + 1.27·23-s + 0.204·24-s − 0.360·25-s − 0.192·27-s − 0.188·28-s + 0.385·29-s + 0.326·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.463522845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463522845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 - 2.07T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 - 1.49T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 0.598T + 71T^{2} \) |
| 73 | \( 1 + 0.423T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 + 4.30T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.055810439916122024631347466921, −6.94833348188028037139201171262, −6.71010639945858413144491950212, −5.91968084555515064686302320993, −5.39144305910616279088443631871, −4.37878818229877189598544530313, −3.48808049531352079047735007952, −2.51705400847658489017934372066, −1.56170618225803408327968276988, −0.74816672361393018811350779005,
0.74816672361393018811350779005, 1.56170618225803408327968276988, 2.51705400847658489017934372066, 3.48808049531352079047735007952, 4.37878818229877189598544530313, 5.39144305910616279088443631871, 5.91968084555515064686302320993, 6.71010639945858413144491950212, 6.94833348188028037139201171262, 8.055810439916122024631347466921
