L(s) = 1 | − 2.97·3-s − 0.794·5-s − 3.63·7-s + 5.87·9-s − 0.674·11-s − 1.02·13-s + 2.36·15-s + 3.65·17-s + 3.51·19-s + 10.8·21-s + 4.04·23-s − 4.36·25-s − 8.54·27-s + 2.95·29-s − 3.73·31-s + 2.00·33-s + 2.88·35-s + 3.30·37-s + 3.06·39-s − 12.0·41-s − 1.05·43-s − 4.66·45-s − 6.46·47-s + 6.24·49-s − 10.8·51-s + 2.27·53-s + 0.535·55-s + ⋯ |
L(s) = 1 | − 1.71·3-s − 0.355·5-s − 1.37·7-s + 1.95·9-s − 0.203·11-s − 0.285·13-s + 0.610·15-s + 0.886·17-s + 0.807·19-s + 2.36·21-s + 0.844·23-s − 0.873·25-s − 1.64·27-s + 0.548·29-s − 0.671·31-s + 0.349·33-s + 0.488·35-s + 0.543·37-s + 0.490·39-s − 1.88·41-s − 0.161·43-s − 0.694·45-s − 0.942·47-s + 0.891·49-s − 1.52·51-s + 0.313·53-s + 0.0721·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4003188875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4003188875\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.97T + 3T^{2} \) |
| 5 | \( 1 + 0.794T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 + 0.674T + 11T^{2} \) |
| 13 | \( 1 + 1.02T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 8.63T + 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.69326692320980853141625919912, −7.29344797047827709818783292185, −6.41114953367858934596406325085, −6.09148052464899628045691499780, −5.22692970859695059587359946284, −4.74708754322668651028439933307, −3.62703379419490656145549566268, −3.03664676288141334463036181606, −1.47252157291671972227317965118, −0.37904395860689514337657560141,
0.37904395860689514337657560141, 1.47252157291671972227317965118, 3.03664676288141334463036181606, 3.62703379419490656145549566268, 4.74708754322668651028439933307, 5.22692970859695059587359946284, 6.09148052464899628045691499780, 6.41114953367858934596406325085, 7.29344797047827709818783292185, 7.69326692320980853141625919912