| L(s) = 1 | + 2-s − 2.89·3-s + 4-s − 5-s − 2.89·6-s − 2.11·7-s + 8-s + 5.35·9-s − 10-s − 4.11·11-s − 2.89·12-s − 4.31·13-s − 2.11·14-s + 2.89·15-s + 16-s + 3.02·17-s + 5.35·18-s + 4.72·19-s − 20-s + 6.10·21-s − 4.11·22-s − 0.00467·23-s − 2.89·24-s + 25-s − 4.31·26-s − 6.82·27-s − 2.11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.66·3-s + 0.5·4-s − 0.447·5-s − 1.18·6-s − 0.798·7-s + 0.353·8-s + 1.78·9-s − 0.316·10-s − 1.23·11-s − 0.834·12-s − 1.19·13-s − 0.564·14-s + 0.746·15-s + 0.250·16-s + 0.733·17-s + 1.26·18-s + 1.08·19-s − 0.223·20-s + 1.33·21-s − 0.876·22-s − 0.000975·23-s − 0.590·24-s + 0.200·25-s − 0.847·26-s − 1.31·27-s − 0.399·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 0.00467T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 37 | \( 1 + 9.55T + 37T^{2} \) |
| 41 | \( 1 + 7.99T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 - 6.12T + 61T^{2} \) |
| 67 | \( 1 - 4.49T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 3.10T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.441T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.16075931906780517783877200399, −6.63947685966842697939075182689, −5.67244546024341603595209532851, −5.31663936143274958330977985734, −4.91982015632868747913061851209, −3.95461763962278681239527617394, −3.17589448021604894736152101233, −2.31742079036287802333179836799, −0.908056910352997855281189561970, 0,
0.908056910352997855281189561970, 2.31742079036287802333179836799, 3.17589448021604894736152101233, 3.95461763962278681239527617394, 4.91982015632868747913061851209, 5.31663936143274958330977985734, 5.67244546024341603595209532851, 6.63947685966842697939075182689, 7.16075931906780517783877200399
