| L(s) = 1 | − 2.06·2-s − 3-s + 2.26·4-s − 0.587·5-s + 2.06·6-s − 1.74·7-s − 0.541·8-s + 9-s + 1.21·10-s + 6.19·11-s − 2.26·12-s + 13-s + 3.59·14-s + 0.587·15-s − 3.40·16-s − 7.08·17-s − 2.06·18-s + 5.77·19-s − 1.32·20-s + 1.74·21-s − 12.7·22-s + 4.47·23-s + 0.541·24-s − 4.65·25-s − 2.06·26-s − 27-s − 3.93·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.13·4-s − 0.262·5-s + 0.842·6-s − 0.657·7-s − 0.191·8-s + 0.333·9-s + 0.383·10-s + 1.86·11-s − 0.653·12-s + 0.277·13-s + 0.960·14-s + 0.151·15-s − 0.851·16-s − 1.71·17-s − 0.486·18-s + 1.32·19-s − 0.297·20-s + 0.379·21-s − 2.72·22-s + 0.932·23-s + 0.110·24-s − 0.930·25-s − 0.404·26-s − 0.192·27-s − 0.744·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 0.587T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 - 0.399T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 7.18T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 7.42T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−8.248226544215623337544200323477, −7.29402116732219793521414164849, −6.73342089845149289770497745226, −6.34858135906526637198462970236, −5.15693587701889257518766948891, −4.16569010569140856312412810404, −3.41865420736141204888776929384, −1.96482121622874948141635901962, −1.10844791533072695412262413180, 0,
1.10844791533072695412262413180, 1.96482121622874948141635901962, 3.41865420736141204888776929384, 4.16569010569140856312412810404, 5.15693587701889257518766948891, 6.34858135906526637198462970236, 6.73342089845149289770497745226, 7.29402116732219793521414164849, 8.248226544215623337544200323477
