| L(s) = 1 | − 0.615·2-s + 2.88·3-s − 1.62·4-s − 1.40·5-s − 1.77·6-s − 0.315·7-s + 2.22·8-s + 5.34·9-s + 0.866·10-s − 4.26·11-s − 4.68·12-s − 13-s + 0.194·14-s − 4.06·15-s + 1.87·16-s − 4.52·17-s − 3.28·18-s + 1.77·19-s + 2.28·20-s − 0.911·21-s + 2.62·22-s − 9.28·23-s + 6.43·24-s − 3.01·25-s + 0.615·26-s + 6.75·27-s + 0.511·28-s + ⋯ |
| L(s) = 1 | − 0.435·2-s + 1.66·3-s − 0.810·4-s − 0.629·5-s − 0.725·6-s − 0.119·7-s + 0.787·8-s + 1.78·9-s + 0.274·10-s − 1.28·11-s − 1.35·12-s − 0.277·13-s + 0.0519·14-s − 1.05·15-s + 0.467·16-s − 1.09·17-s − 0.774·18-s + 0.407·19-s + 0.510·20-s − 0.198·21-s + 0.559·22-s − 1.93·23-s + 1.31·24-s − 0.603·25-s + 0.120·26-s + 1.30·27-s + 0.0967·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.615T + 2T^{2} \) |
| 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 + 0.315T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 + 9.28T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 0.263T + 31T^{2} \) |
| 37 | \( 1 - 9.93T + 37T^{2} \) |
| 41 | \( 1 - 8.44T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 + 7.88T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 - 2.01T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.242207632761590990566678501001, −8.252670458240920885332643543652, −7.87065484505416779556006223123, −7.42917675205600031629252501209, −5.86630262478330621803661026370, −4.51092676904196550626788890844, −4.01814282443674949875868776432, −2.93069331796824267660840405544, −1.95729252183212133041658074709, 0,
1.95729252183212133041658074709, 2.93069331796824267660840405544, 4.01814282443674949875868776432, 4.51092676904196550626788890844, 5.86630262478330621803661026370, 7.42917675205600031629252501209, 7.87065484505416779556006223123, 8.252670458240920885332643543652, 9.242207632761590990566678501001
