| L(s) = 1 | + 3-s − 1.56·5-s + 5.12·7-s + 9-s + 2.43·11-s − 3.56·13-s − 1.56·15-s + 4.68·19-s + 5.12·21-s + 7.56·23-s − 2.56·25-s + 27-s + 7.12·29-s − 8.24·31-s + 2.43·33-s − 8·35-s + 4·37-s − 3.56·39-s + 2.68·41-s − 4.68·43-s − 1.56·45-s − 0.876·47-s + 19.2·49-s + 6·53-s − 3.80·55-s + 4.68·57-s − 13.3·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.698·5-s + 1.93·7-s + 0.333·9-s + 0.735·11-s − 0.987·13-s − 0.403·15-s + 1.07·19-s + 1.11·21-s + 1.57·23-s − 0.512·25-s + 0.192·27-s + 1.32·29-s − 1.48·31-s + 0.424·33-s − 1.35·35-s + 0.657·37-s − 0.570·39-s + 0.419·41-s − 0.714·43-s − 0.232·45-s − 0.127·47-s + 2.74·49-s + 0.824·53-s − 0.513·55-s + 0.620·57-s − 1.74·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.171985463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.171985463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 0.876T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−7.87058724292613022102289505018, −7.43150891023759637831324653602, −6.92759501364971886316774892323, −5.64721637689334511603472487308, −4.88487571505503665471190327248, −4.48749531179977551805582501604, −3.61499116599088114045273950190, −2.72023160314759770117993327027, −1.75844291488568003374331706461, −0.955577207304068598837358652150,
0.955577207304068598837358652150, 1.75844291488568003374331706461, 2.72023160314759770117993327027, 3.61499116599088114045273950190, 4.48749531179977551805582501604, 4.88487571505503665471190327248, 5.64721637689334511603472487308, 6.92759501364971886316774892323, 7.43150891023759637831324653602, 7.87058724292613022102289505018
