| L(s) = 1 | − 2.50·2-s + 1.22·3-s + 4.28·4-s − 3.06·6-s + 0.221·7-s − 5.72·8-s − 1.50·9-s − 0.778·11-s + 5.23·12-s − 5·13-s − 0.556·14-s + 5.79·16-s − 7.07·17-s + 3.77·18-s + 0.271·21-s + 1.95·22-s − 8.07·23-s − 6.99·24-s + 12.5·26-s − 5.50·27-s + 0.950·28-s − 0.221·29-s − 2.50·31-s − 3.06·32-s − 0.950·33-s + 17.7·34-s − 6.45·36-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.705·3-s + 2.14·4-s − 1.25·6-s + 0.0838·7-s − 2.02·8-s − 0.502·9-s − 0.234·11-s + 1.51·12-s − 1.38·13-s − 0.148·14-s + 1.44·16-s − 1.71·17-s + 0.890·18-s + 0.0591·21-s + 0.415·22-s − 1.68·23-s − 1.42·24-s + 2.45·26-s − 1.05·27-s + 0.179·28-s − 0.0412·29-s − 0.450·31-s − 0.541·32-s − 0.165·33-s + 3.04·34-s − 1.07·36-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
| \(L(1)\) |
\(\approx\) |
\(0.2852295753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2852295753\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 1.22T + 3T^{2} \) |
| 7 | \( 1 - 0.221T + 7T^{2} \) |
| 11 | \( 1 + 0.778T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 23 | \( 1 + 8.07T + 23T^{2} \) |
| 29 | \( 1 + 0.221T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 + 7.29T + 43T^{2} \) |
| 47 | \( 1 - 2.79T + 47T^{2} \) |
| 53 | \( 1 - 4.38T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 9.84T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−7.88749469992469980766019401245, −7.43939110794153368303009258934, −6.66988322715731369832033722806, −6.03899863130856749501543558185, −5.01670319311591247777601281698, −4.08448504708748385923883245660, −2.97851127867496390445879703573, −2.23068239143645594429709395066, −1.90775129846922255776135464366, −0.29949680997120040357678833164,
0.29949680997120040357678833164, 1.90775129846922255776135464366, 2.23068239143645594429709395066, 2.97851127867496390445879703573, 4.08448504708748385923883245660, 5.01670319311591247777601281698, 6.03899863130856749501543558185, 6.66988322715731369832033722806, 7.43939110794153368303009258934, 7.88749469992469980766019401245
