| L(s) = 1 | − 0.751·2-s − 0.580·3-s − 1.43·4-s + 0.435·6-s + 0.315·7-s + 2.58·8-s − 2.66·9-s − 4.34·11-s + 0.833·12-s + 4.58·13-s − 0.236·14-s + 0.933·16-s + 0.917·17-s + 2·18-s + 2.76·19-s − 0.182·21-s + 3.26·22-s − 23-s − 1.49·24-s − 3.44·26-s + 3.28·27-s − 0.452·28-s + 7.03·29-s − 0.867·31-s − 5.86·32-s + 2.52·33-s − 0.689·34-s + ⋯ |
| L(s) = 1 | − 0.531·2-s − 0.335·3-s − 0.717·4-s + 0.177·6-s + 0.119·7-s + 0.912·8-s − 0.887·9-s − 1.31·11-s + 0.240·12-s + 1.27·13-s − 0.0632·14-s + 0.233·16-s + 0.222·17-s + 0.471·18-s + 0.634·19-s − 0.0399·21-s + 0.696·22-s − 0.208·23-s − 0.305·24-s − 0.674·26-s + 0.632·27-s − 0.0854·28-s + 1.30·29-s − 0.155·31-s − 1.03·32-s + 0.439·33-s − 0.118·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7258543769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7258543769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.751T + 2T^{2} \) |
| 3 | \( 1 + 0.580T + 3T^{2} \) |
| 7 | \( 1 - 0.315T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 0.917T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 0.867T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 - 9.08T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 7.39T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−10.66151176209710654685675365377, −9.904631978097858792225640145695, −8.807517117588258958022757328398, −8.278266585015634226988482681457, −7.42720485621817451973495191955, −5.95068095470485595961721833205, −5.30641567956163323581022170933, −4.15187581709362657885258688713, −2.78517783335880054939484483933, −0.842244458535962951617433304652,
0.842244458535962951617433304652, 2.78517783335880054939484483933, 4.15187581709362657885258688713, 5.30641567956163323581022170933, 5.95068095470485595961721833205, 7.42720485621817451973495191955, 8.278266585015634226988482681457, 8.807517117588258958022757328398, 9.904631978097858792225640145695, 10.66151176209710654685675365377
