| L(s) = 1 | − 1.41·2-s + 3.19·3-s + 0.0122·4-s − 4.52·6-s + 4.71·7-s + 2.81·8-s + 7.17·9-s − 5.53·11-s + 0.0389·12-s − 0.648·13-s − 6.68·14-s − 4.02·16-s − 1.05·17-s − 10.1·18-s + 7.88·19-s + 15.0·21-s + 7.85·22-s + 3.05·23-s + 8.99·24-s + 0.919·26-s + 13.3·27-s + 0.0576·28-s − 3.58·29-s + 4.18·31-s + 0.0691·32-s − 17.6·33-s + 1.49·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.84·3-s + 0.00611·4-s − 1.84·6-s + 1.78·7-s + 0.996·8-s + 2.39·9-s − 1.66·11-s + 0.0112·12-s − 0.179·13-s − 1.78·14-s − 1.00·16-s − 0.256·17-s − 2.40·18-s + 1.80·19-s + 3.28·21-s + 1.67·22-s + 0.636·23-s + 1.83·24-s + 0.180·26-s + 2.56·27-s + 0.0108·28-s − 0.666·29-s + 0.752·31-s + 0.0122·32-s − 3.07·33-s + 0.257·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047839238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047839238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 + 0.648T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 - 3.05T + 23T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 + 2.09T + 97T^{2} \) |
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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
−9.491024998216676753251525842656, −8.854144424228978640813119722853, −8.113585193897371544482614648993, −7.72351208701288595822764178831, −7.31782393110111902031729811207, −5.10225314876099663178934322471, −4.66727126118180984612365100945, −3.28022654987485053701132760064, −2.22551777840511203067198245778, −1.33614971734622589486744353488,
1.33614971734622589486744353488, 2.22551777840511203067198245778, 3.28022654987485053701132760064, 4.66727126118180984612365100945, 5.10225314876099663178934322471, 7.31782393110111902031729811207, 7.72351208701288595822764178831, 8.113585193897371544482614648993, 8.854144424228978640813119722853, 9.491024998216676753251525842656
