| L(s) = 1 | + 1.17·2-s − 0.616·4-s + 3.14·5-s − 3.07·8-s + 3.70·10-s + 0.773·11-s − 13-s − 2.38·16-s − 5.75·17-s + 1.22·19-s − 1.94·20-s + 0.909·22-s + 2.99·23-s + 4.91·25-s − 1.17·26-s + 2.46·29-s + 6.13·31-s + 3.34·32-s − 6.76·34-s + 4.99·37-s + 1.43·38-s − 9.69·40-s − 2.55·41-s − 2.73·43-s − 0.476·44-s + 3.51·46-s + 5.37·47-s + ⋯ |
| L(s) = 1 | + 0.831·2-s − 0.308·4-s + 1.40·5-s − 1.08·8-s + 1.17·10-s + 0.233·11-s − 0.277·13-s − 0.596·16-s − 1.39·17-s + 0.280·19-s − 0.434·20-s + 0.193·22-s + 0.623·23-s + 0.982·25-s − 0.230·26-s + 0.458·29-s + 1.10·31-s + 0.591·32-s − 1.16·34-s + 0.821·37-s + 0.233·38-s − 1.53·40-s − 0.399·41-s − 0.416·43-s − 0.0718·44-s + 0.518·46-s + 0.783·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265283655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265283655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 - 0.773T + 11T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 - 2.99T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 0.542T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 - 1.26T + 97T^{2} \) |
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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
−8.330781053407755540629480184515, −7.08266807768052354606464084676, −6.45767633932950488412713987783, −5.91988538438232919854733187587, −5.14650410777053655462709631322, −4.64550056822500227159722985399, −3.79781777032719919896746753269, −2.73803075443526072774414710598, −2.18420729495515010390116591407, −0.851841989603352015285704727387,
0.851841989603352015285704727387, 2.18420729495515010390116591407, 2.73803075443526072774414710598, 3.79781777032719919896746753269, 4.64550056822500227159722985399, 5.14650410777053655462709631322, 5.91988538438232919854733187587, 6.45767633932950488412713987783, 7.08266807768052354606464084676, 8.330781053407755540629480184515
