| L(s) = 1 | + 2.05·2-s − 3-s + 2.23·4-s − 1.05·5-s − 2.05·6-s − 7-s + 0.492·8-s + 9-s − 2.18·10-s − 2.23·12-s + 3.80·13-s − 2.05·14-s + 1.05·15-s − 3.46·16-s − 2.56·17-s + 2.05·18-s − 0.180·19-s − 2.37·20-s + 21-s + 0.433·23-s − 0.492·24-s − 3.87·25-s + 7.83·26-s − 27-s − 2.23·28-s − 9.03·29-s + 2.18·30-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.11·4-s − 0.473·5-s − 0.840·6-s − 0.377·7-s + 0.174·8-s + 0.333·9-s − 0.689·10-s − 0.646·12-s + 1.05·13-s − 0.550·14-s + 0.273·15-s − 0.866·16-s − 0.622·17-s + 0.485·18-s − 0.0413·19-s − 0.530·20-s + 0.218·21-s + 0.0904·23-s − 0.100·24-s − 0.775·25-s + 1.53·26-s − 0.192·27-s − 0.423·28-s − 1.67·29-s + 0.398·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 - 0.433T + 23T^{2} \) |
| 29 | \( 1 + 9.03T + 29T^{2} \) |
| 31 | \( 1 + 0.492T + 31T^{2} \) |
| 37 | \( 1 + 0.775T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 - 0.502T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.27T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.493112168971854226151982047756, −7.40568574507502325019133677971, −6.70590129699153478064048787113, −5.94375015051348276902339504745, −5.45593880370570704753244482429, −4.40795916074480952784951393603, −3.87275359263051927920884287790, −3.10073278778123050233274307420, −1.79927877449918316164283996618, 0,
1.79927877449918316164283996618, 3.10073278778123050233274307420, 3.87275359263051927920884287790, 4.40795916074480952784951393603, 5.45593880370570704753244482429, 5.94375015051348276902339504745, 6.70590129699153478064048787113, 7.40568574507502325019133677971, 8.493112168971854226151982047756
