| L(s) = 1 | − 2.17·2-s − 0.539·3-s + 2.70·4-s + 1.17·6-s + 4.87·7-s − 1.53·8-s − 2.70·9-s − 3.17·11-s − 1.46·12-s − 2.63·13-s − 10.5·14-s − 2.07·16-s + 5.87·18-s − 1.07·19-s − 2.63·21-s + 6.87·22-s − 5.21·23-s + 0.829·24-s + 5.70·26-s + 3.07·27-s + 13.2·28-s + 2.92·29-s − 4.09·31-s + 7.58·32-s + 1.70·33-s − 7.34·36-s + 5.26·37-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.311·3-s + 1.35·4-s + 0.477·6-s + 1.84·7-s − 0.544·8-s − 0.903·9-s − 0.955·11-s − 0.421·12-s − 0.729·13-s − 2.82·14-s − 0.519·16-s + 1.38·18-s − 0.247·19-s − 0.574·21-s + 1.46·22-s − 1.08·23-s + 0.169·24-s + 1.11·26-s + 0.592·27-s + 2.49·28-s + 0.542·29-s − 0.734·31-s + 1.34·32-s + 0.297·33-s − 1.22·36-s + 0.865·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 + 0.539T + 3T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.06T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 7.15T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81926311829242775538073375075, −7.31752567893864269880196510177, −6.32953661588520188037311666911, −5.39411866618443275555028220215, −4.97196452208940990764232721823, −4.06151011461869105488551419557, −2.50002870165130427363582329632, −2.15242520493678666151270192096, −1.04180897518284733649468456931, 0,
1.04180897518284733649468456931, 2.15242520493678666151270192096, 2.50002870165130427363582329632, 4.06151011461869105488551419557, 4.97196452208940990764232721823, 5.39411866618443275555028220215, 6.32953661588520188037311666911, 7.31752567893864269880196510177, 7.81926311829242775538073375075
