L(s) = 1 | − 3-s − 0.0590·5-s − 1.12·7-s + 9-s − 1.48·11-s − 2.44·13-s + 0.0590·15-s + 1.10·17-s − 2.20·19-s + 1.12·21-s + 7.94·23-s − 4.99·25-s − 27-s − 2.34·29-s + 8.59·31-s + 1.48·33-s + 0.0666·35-s + 9.57·37-s + 2.44·39-s + 6.13·41-s + 0.543·43-s − 0.0590·45-s − 7.76·47-s − 5.72·49-s − 1.10·51-s − 7.73·53-s + 0.0876·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.0264·5-s − 0.426·7-s + 0.333·9-s − 0.447·11-s − 0.678·13-s + 0.0152·15-s + 0.267·17-s − 0.506·19-s + 0.246·21-s + 1.65·23-s − 0.999·25-s − 0.192·27-s − 0.435·29-s + 1.54·31-s + 0.258·33-s + 0.0112·35-s + 1.57·37-s + 0.391·39-s + 0.957·41-s + 0.0828·43-s − 0.00880·45-s − 1.13·47-s − 0.817·49-s − 0.154·51-s − 1.06·53-s + 0.0118·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 - T \) |
good | 5 | \( 1 + 0.0590T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 - 8.59T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 - 0.543T + 43T^{2} \) |
| 47 | \( 1 + 7.76T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 + 6.40T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 6.03T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 + 4.79T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.72918062195495657401944447257, −6.91296960669026816545409845104, −6.32659853698038905299962048616, −5.59461004237933397927528078954, −4.84440631207107561354937531376, −4.22218390617666990062029422791, −3.12707748430831435265785985462, −2.42387875235785624661632302287, −1.15100260388262728991956245665, 0,
1.15100260388262728991956245665, 2.42387875235785624661632302287, 3.12707748430831435265785985462, 4.22218390617666990062029422791, 4.84440631207107561354937531376, 5.59461004237933397927528078954, 6.32659853698038905299962048616, 6.91296960669026816545409845104, 7.72918062195495657401944447257