| L(s) = 1 | − 4.83·2-s + 3·3-s + 15.4·4-s + 21.1·5-s − 14.5·6-s + 16.2·7-s − 35.8·8-s + 9·9-s − 102.·10-s + 30.7·11-s + 46.2·12-s − 78.7·14-s + 63.5·15-s + 49.9·16-s + 46.2·17-s − 43.5·18-s + 144.·19-s + 326.·20-s + 48.8·21-s − 148.·22-s + 8.38·23-s − 107.·24-s + 324.·25-s + 27·27-s + 250.·28-s − 242.·29-s − 307.·30-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.92·4-s + 1.89·5-s − 0.987·6-s + 0.879·7-s − 1.58·8-s + 0.333·9-s − 3.24·10-s + 0.842·11-s + 1.11·12-s − 1.50·14-s + 1.09·15-s + 0.781·16-s + 0.659·17-s − 0.570·18-s + 1.74·19-s + 3.65·20-s + 0.507·21-s − 1.44·22-s + 0.0759·23-s − 0.913·24-s + 2.59·25-s + 0.192·27-s + 1.69·28-s − 1.55·29-s − 1.87·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.975928786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975928786\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.83T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 - 30.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 46.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.38T + 1.21e4T^{2} \) |
| 29 | \( 1 + 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 49.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 348.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 48.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 984.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 487.T + 9.12e5T^{2} \) |
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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
−10.01302722680998578743603635966, −9.533991889329287485883968564777, −9.015272700522131117629562163208, −8.020662569089521234884803754439, −7.17429139590880076852285152236, −6.16297599270322822999854703429, −5.09703954685105148207997446783, −3.01785232433449180035147183958, −1.70185259895792534204976877913, −1.33788886003258947017753531651,
1.33788886003258947017753531651, 1.70185259895792534204976877913, 3.01785232433449180035147183958, 5.09703954685105148207997446783, 6.16297599270322822999854703429, 7.17429139590880076852285152236, 8.020662569089521234884803754439, 9.015272700522131117629562163208, 9.533991889329287485883968564777, 10.01302722680998578743603635966
