L(s) = 1 | + 2-s + 1.16·3-s + 4-s − 1.55·5-s + 1.16·6-s + 7-s + 8-s − 1.64·9-s − 1.55·10-s + 0.834·11-s + 1.16·12-s − 5.35·13-s + 14-s − 1.80·15-s + 16-s + 3.47·17-s − 1.64·18-s − 1.55·20-s + 1.16·21-s + 0.834·22-s + 8.74·23-s + 1.16·24-s − 2.59·25-s − 5.35·26-s − 5.41·27-s + 28-s + 9.23·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.672·3-s + 0.5·4-s − 0.693·5-s + 0.475·6-s + 0.377·7-s + 0.353·8-s − 0.547·9-s − 0.490·10-s + 0.251·11-s + 0.336·12-s − 1.48·13-s + 0.267·14-s − 0.466·15-s + 0.250·16-s + 0.842·17-s − 0.386·18-s − 0.346·20-s + 0.254·21-s + 0.177·22-s + 1.82·23-s + 0.237·24-s − 0.519·25-s − 1.05·26-s − 1.04·27-s + 0.188·28-s + 1.71·29-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
\(L(1)\) |
\(\approx\) |
\(3.425651264\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.425651264\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 11 | \( 1 - 0.834T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 23 | \( 1 - 8.74T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 + 9.16T + 61T^{2} \) |
| 67 | \( 1 + 0.309T + 67T^{2} \) |
| 71 | \( 1 + 0.734T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 1.49T + 97T^{2} \) |
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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
−8.054775014746248969844974311835, −7.44532513194593665862714596897, −7.07072860512376673538083992586, −5.85721833904877168163422505961, −5.24753211375210150201754224058, −4.45339043375790289343724553403, −3.73811637394497289747262399706, −2.84506026772308500185522819418, −2.35743463804795524044751076421, −0.876370305860192551768278035886,
0.876370305860192551768278035886, 2.35743463804795524044751076421, 2.84506026772308500185522819418, 3.73811637394497289747262399706, 4.45339043375790289343724553403, 5.24753211375210150201754224058, 5.85721833904877168163422505961, 7.07072860512376673538083992586, 7.44532513194593665862714596897, 8.054775014746248969844974311835