L(s) = 1 | − 2-s + 2.59·3-s + 4-s + 3.85·5-s − 2.59·6-s + 7-s − 8-s + 3.70·9-s − 3.85·10-s + 5.26·11-s + 2.59·12-s + 2.42·13-s − 14-s + 9.98·15-s + 16-s − 3.44·17-s − 3.70·18-s + 3.85·20-s + 2.59·21-s − 5.26·22-s − 5.54·23-s − 2.59·24-s + 9.86·25-s − 2.42·26-s + 1.83·27-s + 28-s − 9.13·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.49·3-s + 0.5·4-s + 1.72·5-s − 1.05·6-s + 0.377·7-s − 0.353·8-s + 1.23·9-s − 1.21·10-s + 1.58·11-s + 0.747·12-s + 0.673·13-s − 0.267·14-s + 2.57·15-s + 0.250·16-s − 0.834·17-s − 0.874·18-s + 0.861·20-s + 0.565·21-s − 1.12·22-s − 1.15·23-s − 0.528·24-s + 1.97·25-s − 0.476·26-s + 0.353·27-s + 0.188·28-s − 1.69·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.060198059\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.060198059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.59T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 + 9.13T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 0.133T + 37T^{2} \) |
| 41 | \( 1 - 6.10T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 - 2.28T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 - 6.81T + 59T^{2} \) |
| 61 | \( 1 + 1.44T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 - 9.83T + 73T^{2} \) |
| 79 | \( 1 + 0.632T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 1.58T + 97T^{2} \) |
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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
−8.690261200692748530111565049763, −7.59772800409604809927116940567, −6.99884474384701548919163165888, −6.08371510502231905919670565882, −5.68919604565261845733702902710, −4.21393603350359542965469009276, −3.61941129485597040489987455028, −2.41430379354158142486526244007, −1.95264121133363215172493764349, −1.30601609936167490332960647576,
1.30601609936167490332960647576, 1.95264121133363215172493764349, 2.41430379354158142486526244007, 3.61941129485597040489987455028, 4.21393603350359542965469009276, 5.68919604565261845733702902710, 6.08371510502231905919670565882, 6.99884474384701548919163165888, 7.59772800409604809927116940567, 8.690261200692748530111565049763