| L(s) = 1 | − 1.70·3-s + 3.97·5-s + 7-s − 0.0942·9-s − 5.17·11-s + 0.739·13-s − 6.77·15-s − 7.56·17-s − 8.53·19-s − 1.70·21-s − 23-s + 10.7·25-s + 5.27·27-s − 8.10·29-s − 4.71·31-s + 8.81·33-s + 3.97·35-s + 6.41·37-s − 1.26·39-s + 2.19·41-s + 7.47·43-s − 0.374·45-s − 0.295·47-s + 49-s + 12.9·51-s − 8.81·53-s − 20.5·55-s + ⋯ |
| L(s) = 1 | − 0.984·3-s + 1.77·5-s + 0.377·7-s − 0.0314·9-s − 1.55·11-s + 0.205·13-s − 1.74·15-s − 1.83·17-s − 1.95·19-s − 0.371·21-s − 0.208·23-s + 2.15·25-s + 1.01·27-s − 1.50·29-s − 0.846·31-s + 1.53·33-s + 0.671·35-s + 1.05·37-s − 0.201·39-s + 0.343·41-s + 1.13·43-s − 0.0558·45-s − 0.0430·47-s + 0.142·49-s + 1.80·51-s − 1.21·53-s − 2.77·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 - 0.739T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 + 8.53T + 19T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 + 4.71T + 31T^{2} \) |
| 37 | \( 1 - 6.41T + 37T^{2} \) |
| 41 | \( 1 - 2.19T + 41T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 + 0.295T + 47T^{2} \) |
| 53 | \( 1 + 8.81T + 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 + 0.444T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−9.241619905802126152407804938626, −8.678789988242428880911694191857, −7.51555848506804893049342953119, −6.27292371752521030566279221506, −6.05787688056212188753257813973, −5.17757629670689852078872385040, −4.46352674535055621436149795127, −2.53361440455487334364498684370, −1.94167769617543934429644677324, 0,
1.94167769617543934429644677324, 2.53361440455487334364498684370, 4.46352674535055621436149795127, 5.17757629670689852078872385040, 6.05787688056212188753257813973, 6.27292371752521030566279221506, 7.51555848506804893049342953119, 8.678789988242428880911694191857, 9.241619905802126152407804938626
