| L(s) = 1 | − 2.31·2-s + 2.90·3-s + 3.35·4-s − 6.72·6-s + 2.34·7-s − 3.14·8-s + 5.43·9-s − 3.45·11-s + 9.74·12-s − 3.29·13-s − 5.43·14-s + 0.554·16-s + 1.38·17-s − 12.5·18-s + 6.81·21-s + 7.98·22-s − 8.90·23-s − 9.11·24-s + 7.61·26-s + 7.06·27-s + 7.87·28-s + 6.28·29-s − 7.39·31-s + 4.99·32-s − 10.0·33-s − 3.20·34-s + 18.2·36-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.67·3-s + 1.67·4-s − 2.74·6-s + 0.886·7-s − 1.11·8-s + 1.81·9-s − 1.04·11-s + 2.81·12-s − 0.912·13-s − 1.45·14-s + 0.138·16-s + 0.335·17-s − 2.96·18-s + 1.48·21-s + 1.70·22-s − 1.85·23-s − 1.86·24-s + 1.49·26-s + 1.35·27-s + 1.48·28-s + 1.16·29-s − 1.32·31-s + 0.883·32-s − 1.74·33-s − 0.548·34-s + 3.03·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 23 | \( 1 + 8.90T + 23T^{2} \) |
| 29 | \( 1 - 6.28T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 0.650T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 - 2.46T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 6.88T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 1.84T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 7.61T + 97T^{2} \) |
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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
−7.77487228849617900922744736949, −7.33171816509220092604648800012, −6.45153294166432356973409742254, −5.27021586676946562875783011159, −4.49466386798673831273185494035, −3.53393454172048971581132017963, −2.55236169795421664945188357194, −2.14813474532044611602592624204, −1.41650539008234798568828215753, 0,
1.41650539008234798568828215753, 2.14813474532044611602592624204, 2.55236169795421664945188357194, 3.53393454172048971581132017963, 4.49466386798673831273185494035, 5.27021586676946562875783011159, 6.45153294166432356973409742254, 7.33171816509220092604648800012, 7.77487228849617900922744736949
