| L(s) = 1 | − 2-s − 0.519·3-s + 4-s + 0.519·6-s − 4.76·7-s − 8-s − 2.72·9-s + 0.960·11-s − 0.519·12-s − 2.24·13-s + 4.76·14-s + 16-s + 0.249·17-s + 2.72·18-s + 19-s + 2.48·21-s − 0.960·22-s + 9.01·23-s + 0.519·24-s + 2.24·26-s + 2.97·27-s − 4.76·28-s + 6.24·29-s + 2.96·31-s − 32-s − 0.499·33-s − 0.249·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.300·3-s + 0.5·4-s + 0.212·6-s − 1.80·7-s − 0.353·8-s − 0.909·9-s + 0.289·11-s − 0.150·12-s − 0.623·13-s + 1.27·14-s + 0.250·16-s + 0.0605·17-s + 0.643·18-s + 0.229·19-s + 0.541·21-s − 0.204·22-s + 1.88·23-s + 0.106·24-s + 0.441·26-s + 0.573·27-s − 0.901·28-s + 1.16·29-s + 0.531·31-s − 0.176·32-s − 0.0868·33-s − 0.0428·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6304550206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6304550206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.519T + 3T^{2} \) |
| 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 - 0.960T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 0.249T + 17T^{2} \) |
| 23 | \( 1 - 9.01T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 0.0399T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 + 5.72T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−9.948091539685308432016879974426, −9.234210075517424403643555650970, −8.639001068839464170886260170496, −7.41217508186696347892222880658, −6.62475566274911169556962867734, −6.04700981419646254201216884446, −4.90734770671744497980694283616, −3.30471999333405805959327841466, −2.70461780836463595661692656779, −0.67496779432486798073596660042,
0.67496779432486798073596660042, 2.70461780836463595661692656779, 3.30471999333405805959327841466, 4.90734770671744497980694283616, 6.04700981419646254201216884446, 6.62475566274911169556962867734, 7.41217508186696347892222880658, 8.639001068839464170886260170496, 9.234210075517424403643555650970, 9.948091539685308432016879974426
