| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 20.4·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 40.8·10-s − 43.5·11-s − 12·12-s + 11.7·13-s + 14·14-s + 61.2·15-s + 16·16-s − 113.·17-s + 18·18-s − 19·19-s − 81.6·20-s − 21·21-s − 87.0·22-s − 37.6·23-s − 24·24-s + 292.·25-s + 23.4·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.82·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.29·10-s − 1.19·11-s − 0.288·12-s + 0.249·13-s + 0.267·14-s + 1.05·15-s + 0.250·16-s − 1.62·17-s + 0.235·18-s − 0.229·19-s − 0.913·20-s − 0.218·21-s − 0.843·22-s − 0.341·23-s − 0.204·24-s + 2.33·25-s + 0.176·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.230871370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230871370\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 20.4T + 125T^{2} \) |
| 11 | \( 1 + 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 37.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 157.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 552.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 121.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 161.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 610.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 515.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 690.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−10.43535486013534106555321993596, −8.677088430028043317948165791905, −8.063124008421635705826147354538, −7.20173740525841738296051629946, −6.44396387476244723685381330613, −5.09795288175227606386384801871, −4.52105874475567123573529099468, −3.67433502782643719414363700239, −2.42152003844069539737092777190, −0.55315562759696659346261061675,
0.55315562759696659346261061675, 2.42152003844069539737092777190, 3.67433502782643719414363700239, 4.52105874475567123573529099468, 5.09795288175227606386384801871, 6.44396387476244723685381330613, 7.20173740525841738296051629946, 8.063124008421635705826147354538, 8.677088430028043317948165791905, 10.43535486013534106555321993596
