| L(s) = 1 | + 3.11·2-s − 3·3-s + 1.69·4-s + 13.7·5-s − 9.34·6-s + 11.2·7-s − 19.6·8-s + 9·9-s + 42.8·10-s + 20.1·11-s − 5.07·12-s + 9.24·13-s + 34.9·14-s − 41.2·15-s − 74.6·16-s + 127.·17-s + 28.0·18-s + 98.8·19-s + 23.3·20-s − 33.7·21-s + 62.6·22-s − 17.1·23-s + 58.9·24-s + 64.4·25-s + 28.7·26-s − 27·27-s + 19.0·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.211·4-s + 1.23·5-s − 0.635·6-s + 0.606·7-s − 0.867·8-s + 0.333·9-s + 1.35·10-s + 0.551·11-s − 0.122·12-s + 0.197·13-s + 0.667·14-s − 0.710·15-s − 1.16·16-s + 1.82·17-s + 0.366·18-s + 1.19·19-s + 0.260·20-s − 0.350·21-s + 0.607·22-s − 0.155·23-s + 0.501·24-s + 0.515·25-s + 0.217·26-s − 0.192·27-s + 0.128·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.103882794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.103882794\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 67 | \( 1 + 67T \) |
| good | 2 | \( 1 - 3.11T + 8T^{2} \) |
| 5 | \( 1 - 13.7T + 125T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 - 20.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.24T + 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 301.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 365.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 887.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 405.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 429.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 486.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 66.5T + 9.12e5T^{2} \) |
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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
−12.08653958526616398878135035601, −11.40351771877102923140297007771, −9.986547775292678271926072967523, −9.337066527224229593721604577620, −7.74772004430414524117393789472, −6.19669800937459983709763973999, −5.62921674394719784747437493158, −4.70199437257010689463673031791, −3.23864445734698344126175567773, −1.40883012043942386201571903980,
1.40883012043942386201571903980, 3.23864445734698344126175567773, 4.70199437257010689463673031791, 5.62921674394719784747437493158, 6.19669800937459983709763973999, 7.74772004430414524117393789472, 9.337066527224229593721604577620, 9.986547775292678271926072967523, 11.40351771877102923140297007771, 12.08653958526616398878135035601
