| L(s) = 1 | − 2.37·3-s + 5-s + 4.57·7-s + 2.66·9-s + 5.84·11-s + 2.86·13-s − 2.37·15-s − 6.90·17-s − 2.24·19-s − 10.8·21-s − 0.981·23-s + 25-s + 0.798·27-s + 29-s + 4.18·31-s − 13.9·33-s + 4.57·35-s + 2.31·37-s − 6.81·39-s − 7.15·41-s − 4.90·43-s + 2.66·45-s + 6.85·47-s + 13.9·49-s + 16.4·51-s − 6.06·53-s + 5.84·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s + 0.447·5-s + 1.73·7-s + 0.888·9-s + 1.76·11-s + 0.794·13-s − 0.614·15-s − 1.67·17-s − 0.514·19-s − 2.37·21-s − 0.204·23-s + 0.200·25-s + 0.153·27-s + 0.185·29-s + 0.751·31-s − 2.42·33-s + 0.773·35-s + 0.380·37-s − 1.09·39-s − 1.11·41-s − 0.748·43-s + 0.397·45-s + 0.999·47-s + 1.99·49-s + 2.30·51-s − 0.833·53-s + 0.788·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477272815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477272815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 - 5.84T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 0.981T + 23T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 - 6.85T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.30T + 67T^{2} \) |
| 71 | \( 1 - 9.51T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−9.990738479409832739612090624503, −8.770056319760676477320354069320, −8.413592644719380525798106885673, −6.88900402700430252443189005328, −6.45416110907705669585843371809, −5.57226137701354809297742460387, −4.66250519650252249717173174536, −4.08062410984427401847682360068, −2.00443879036301404979573102681, −1.08294990422208240121632186307,
1.08294990422208240121632186307, 2.00443879036301404979573102681, 4.08062410984427401847682360068, 4.66250519650252249717173174536, 5.57226137701354809297742460387, 6.45416110907705669585843371809, 6.88900402700430252443189005328, 8.413592644719380525798106885673, 8.770056319760676477320354069320, 9.990738479409832739612090624503
