L(s) = 1 | − 2-s − 2.73·3-s + 4-s − 3.97·5-s + 2.73·6-s + 4.88·7-s − 8-s + 4.48·9-s + 3.97·10-s − 1.47·11-s − 2.73·12-s + 0.404·13-s − 4.88·14-s + 10.8·15-s + 16-s + 4.76·17-s − 4.48·18-s + 2.11·19-s − 3.97·20-s − 13.3·21-s + 1.47·22-s + 0.955·23-s + 2.73·24-s + 10.8·25-s − 0.404·26-s − 4.06·27-s + 4.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.57·3-s + 0.5·4-s − 1.77·5-s + 1.11·6-s + 1.84·7-s − 0.353·8-s + 1.49·9-s + 1.25·10-s − 0.444·11-s − 0.789·12-s + 0.112·13-s − 1.30·14-s + 2.80·15-s + 0.250·16-s + 1.15·17-s − 1.05·18-s + 0.484·19-s − 0.889·20-s − 2.91·21-s + 0.314·22-s + 0.199·23-s + 0.558·24-s + 2.16·25-s − 0.0793·26-s − 0.781·27-s + 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6934274986\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6934274986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 4.88T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 - 0.404T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 0.955T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 - 0.277T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.25T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 9.01T + 73T^{2} \) |
| 79 | \( 1 - 0.856T + 79T^{2} \) |
| 83 | \( 1 - 0.687T + 83T^{2} \) |
| 89 | \( 1 - 4.02T + 89T^{2} \) |
| 97 | \( 1 + 6.53T + 97T^{2} \) |
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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
−8.158532663206612825142967885506, −7.69432396379107971232376307382, −7.31854372253613977429173378019, −6.26950426649457546932545900192, −5.36842561420732259777971125356, −4.75676299006622980924061504284, −4.18258095175379457682871203109, −2.92605560590971447267099157112, −1.29905142855041597960543548675, −0.67762019818223374895559694715,
0.67762019818223374895559694715, 1.29905142855041597960543548675, 2.92605560590971447267099157112, 4.18258095175379457682871203109, 4.75676299006622980924061504284, 5.36842561420732259777971125356, 6.26950426649457546932545900192, 7.31854372253613977429173378019, 7.69432396379107971232376307382, 8.158532663206612825142967885506