| L(s) = 1 | + 4.84·2-s − 9.58·3-s + 15.4·4-s − 46.3·6-s + 7·7-s + 36.0·8-s + 64.7·9-s + 62.1·11-s − 147.·12-s + 14.0·13-s + 33.8·14-s + 50.9·16-s + 63.5·17-s + 313.·18-s + 48.7·19-s − 67.0·21-s + 301.·22-s − 99.3·23-s − 345.·24-s + 68.2·26-s − 362.·27-s + 108.·28-s − 69.0·29-s − 9.68·31-s − 41.7·32-s − 595.·33-s + 307.·34-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 1.84·3-s + 1.93·4-s − 3.15·6-s + 0.377·7-s + 1.59·8-s + 2.39·9-s + 1.70·11-s − 3.55·12-s + 0.300·13-s + 0.646·14-s + 0.795·16-s + 0.906·17-s + 4.10·18-s + 0.588·19-s − 0.696·21-s + 2.91·22-s − 0.900·23-s − 2.93·24-s + 0.514·26-s − 2.58·27-s + 0.729·28-s − 0.442·29-s − 0.0561·31-s − 0.230·32-s − 3.14·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.965741682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965741682\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 4.84T + 8T^{2} \) |
| 3 | \( 1 + 9.58T + 27T^{2} \) |
| 11 | \( 1 - 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.68T + 2.97e4T^{2} \) |
| 37 | \( 1 - 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 305.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 504.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 513.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.86e3T + 9.12e5T^{2} \) |
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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
−12.10037201381193873882679650320, −11.65505903800325254953813146299, −10.98230026189314591039648801696, −9.665859362322318616591747280309, −7.40112779910478852028997592301, −6.30302105881551788717011409308, −5.79567778775109122752182892771, −4.68188531151766533414820375917, −3.80684917722860389696762232536, −1.35796682910238191542612485570,
1.35796682910238191542612485570, 3.80684917722860389696762232536, 4.68188531151766533414820375917, 5.79567778775109122752182892771, 6.30302105881551788717011409308, 7.40112779910478852028997592301, 9.665859362322318616591747280309, 10.98230026189314591039648801696, 11.65505903800325254953813146299, 12.10037201381193873882679650320
