| L(s) = 1 | − 2-s − 0.724·3-s + 4-s − 3.58·5-s + 0.724·6-s − 1.97·7-s − 8-s − 2.47·9-s + 3.58·10-s + 11-s − 0.724·12-s + 3.39·13-s + 1.97·14-s + 2.59·15-s + 16-s − 3.86·17-s + 2.47·18-s − 3.58·20-s + 1.43·21-s − 22-s − 7.14·23-s + 0.724·24-s + 7.82·25-s − 3.39·26-s + 3.96·27-s − 1.97·28-s + 4.29·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.418·3-s + 0.5·4-s − 1.60·5-s + 0.295·6-s − 0.747·7-s − 0.353·8-s − 0.825·9-s + 1.13·10-s + 0.301·11-s − 0.209·12-s + 0.942·13-s + 0.528·14-s + 0.669·15-s + 0.250·16-s − 0.936·17-s + 0.583·18-s − 0.800·20-s + 0.312·21-s − 0.213·22-s − 1.48·23-s + 0.147·24-s + 1.56·25-s − 0.666·26-s + 0.763·27-s − 0.373·28-s + 0.797·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.724T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 + 1.97T + 7T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 + 3.91T + 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 - 5.06T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 + 0.601T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 + 0.207T + 97T^{2} \) |
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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
−7.54703048745563222506905669264, −6.87681802303078001218708016395, −6.25652413662864791549219473196, −5.65106419109199586985215206838, −4.49026896881364477824527732160, −3.79850698386422243547197371369, −3.22821720910100822061827947704, −2.19552366235413824568918172330, −0.76816880500018772480896478440, 0,
0.76816880500018772480896478440, 2.19552366235413824568918172330, 3.22821720910100822061827947704, 3.79850698386422243547197371369, 4.49026896881364477824527732160, 5.65106419109199586985215206838, 6.25652413662864791549219473196, 6.87681802303078001218708016395, 7.54703048745563222506905669264
