| L(s) = 1 | + 1.30·3-s − 2.99·5-s − 3.88·7-s − 1.30·9-s − 4.33·11-s + 13-s − 3.89·15-s − 2.80·17-s + 3.12·19-s − 5.05·21-s − 8.23·23-s + 3.95·25-s − 5.60·27-s − 29-s + 6.16·31-s − 5.63·33-s + 11.6·35-s − 5.85·37-s + 1.30·39-s − 0.970·41-s − 1.37·43-s + 3.91·45-s − 8.52·47-s + 8.11·49-s − 3.64·51-s − 5.70·53-s + 12.9·55-s + ⋯ |
| L(s) = 1 | + 0.750·3-s − 1.33·5-s − 1.46·7-s − 0.436·9-s − 1.30·11-s + 0.277·13-s − 1.00·15-s − 0.679·17-s + 0.716·19-s − 1.10·21-s − 1.71·23-s + 0.790·25-s − 1.07·27-s − 0.185·29-s + 1.10·31-s − 0.980·33-s + 1.96·35-s − 0.961·37-s + 0.208·39-s − 0.151·41-s − 0.209·43-s + 0.583·45-s − 1.24·47-s + 1.15·49-s − 0.510·51-s − 0.783·53-s + 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3433308835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3433308835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 + 0.970T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 + 5.70T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 6.01T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.088948598874188404668641150742, −7.61877180689676987041963787653, −6.72833137738360896403241108751, −6.07151155168064791288897455899, −5.15533024239984750578692537545, −4.18638474865742293873727320962, −3.40526194876556048300788227199, −3.07548051450105321462980248488, −2.14613878752981366426203263806, −0.27130323967605840259674849353,
0.27130323967605840259674849353, 2.14613878752981366426203263806, 3.07548051450105321462980248488, 3.40526194876556048300788227199, 4.18638474865742293873727320962, 5.15533024239984750578692537545, 6.07151155168064791288897455899, 6.72833137738360896403241108751, 7.61877180689676987041963787653, 8.088948598874188404668641150742
