| L(s) = 1 | − 0.441·2-s + 3-s − 1.80·4-s − 4.28·5-s − 0.441·6-s + 3.18·7-s + 1.67·8-s + 9-s + 1.89·10-s − 1.80·12-s − 2.06·13-s − 1.40·14-s − 4.28·15-s + 2.86·16-s + 17-s − 0.441·18-s − 6.57·19-s + 7.74·20-s + 3.18·21-s − 4.53·23-s + 1.67·24-s + 13.3·25-s + 0.913·26-s + 27-s − 5.74·28-s + 2.71·29-s + 1.89·30-s + ⋯ |
| L(s) = 1 | − 0.312·2-s + 0.577·3-s − 0.902·4-s − 1.91·5-s − 0.180·6-s + 1.20·7-s + 0.593·8-s + 0.333·9-s + 0.598·10-s − 0.521·12-s − 0.573·13-s − 0.375·14-s − 1.10·15-s + 0.717·16-s + 0.242·17-s − 0.104·18-s − 1.50·19-s + 1.73·20-s + 0.693·21-s − 0.945·23-s + 0.342·24-s + 2.67·25-s + 0.179·26-s + 0.192·27-s − 1.08·28-s + 0.503·29-s + 0.345·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.441T + 2T^{2} \) |
| 5 | \( 1 + 4.28T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + 8.44T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 8.40T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 5.25T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 - 7.61T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 + 0.924T + 97T^{2} \) |
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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
−8.018385717649307425683077419323, −7.38316038148848512996510323061, −6.55128165099057474038121247473, −5.17868348088454047696193262370, −4.49443393595356584523476074581, −4.22465251006672304006573815909, −3.43715023779559743170344327782, −2.31972894576000714757123748669, −1.07627109087777638648245068468, 0,
1.07627109087777638648245068468, 2.31972894576000714757123748669, 3.43715023779559743170344327782, 4.22465251006672304006573815909, 4.49443393595356584523476074581, 5.17868348088454047696193262370, 6.55128165099057474038121247473, 7.38316038148848512996510323061, 8.018385717649307425683077419323
