| L(s) = 1 | − 2.31·2-s + 2.09·3-s + 3.36·4-s − 4.84·6-s + 2.98·7-s − 3.16·8-s + 1.37·9-s − 2.06·11-s + 7.03·12-s − 2.80·13-s − 6.92·14-s + 0.597·16-s − 8.15·17-s − 3.17·18-s + 6.25·21-s + 4.78·22-s + 4.41·23-s − 6.61·24-s + 6.50·26-s − 3.40·27-s + 10.0·28-s + 8.16·29-s − 5.00·31-s + 4.94·32-s − 4.31·33-s + 18.8·34-s + 4.61·36-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.20·3-s + 1.68·4-s − 1.97·6-s + 1.13·7-s − 1.11·8-s + 0.456·9-s − 0.622·11-s + 2.03·12-s − 0.778·13-s − 1.85·14-s + 0.149·16-s − 1.97·17-s − 0.748·18-s + 1.36·21-s + 1.01·22-s + 0.921·23-s − 1.35·24-s + 1.27·26-s − 0.655·27-s + 1.90·28-s + 1.51·29-s − 0.899·31-s + 0.873·32-s − 0.751·33-s + 3.23·34-s + 0.768·36-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
| \(L(1)\) |
\(\approx\) |
\(1.293341087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.293341087\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.09T + 3T^{2} \) |
| 7 | \( 1 - 2.98T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 2.80T + 13T^{2} \) |
| 17 | \( 1 + 8.15T + 17T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 5.00T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 + 0.799T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 - 0.560T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.999796555381934077181980243390, −7.29676707444262328395039815401, −6.99738968155659180220804813142, −5.86172101913703193671653313479, −4.77974818164072748010556245582, −4.28146784814898549945674912653, −2.88220815025909043353791259126, −2.37438043812554502682762667428, −1.82616071094252632215567003060, −0.64622471014230861823581060239,
0.64622471014230861823581060239, 1.82616071094252632215567003060, 2.37438043812554502682762667428, 2.88220815025909043353791259126, 4.28146784814898549945674912653, 4.77974818164072748010556245582, 5.86172101913703193671653313479, 6.99738968155659180220804813142, 7.29676707444262328395039815401, 7.999796555381934077181980243390
