L(s) = 1 | − 1.30·3-s + 15.1·5-s − 22.0·7-s − 25.3·9-s + 46.1·11-s + 57.4·13-s − 19.6·15-s − 52.7·17-s + 45.2·19-s + 28.7·21-s + 80.1·23-s + 103.·25-s + 68.0·27-s + 29·29-s − 47.9·31-s − 60.0·33-s − 333.·35-s + 91.2·37-s − 74.7·39-s + 252.·41-s − 480.·43-s − 382.·45-s − 551.·47-s + 145.·49-s + 68.6·51-s + 263.·53-s + 696.·55-s + ⋯ |
L(s) = 1 | − 0.250·3-s + 1.35·5-s − 1.19·7-s − 0.937·9-s + 1.26·11-s + 1.22·13-s − 0.338·15-s − 0.751·17-s + 0.545·19-s + 0.298·21-s + 0.726·23-s + 0.824·25-s + 0.485·27-s + 0.185·29-s − 0.277·31-s − 0.316·33-s − 1.61·35-s + 0.405·37-s − 0.307·39-s + 0.961·41-s − 1.70·43-s − 1.26·45-s − 1.71·47-s + 0.423·49-s + 0.188·51-s + 0.682·53-s + 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.357310618\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.357310618\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
good | 3 | \( 1 + 1.30T + 27T^{2} \) |
| 5 | \( 1 - 15.1T + 125T^{2} \) |
| 7 | \( 1 + 22.0T + 343T^{2} \) |
| 11 | \( 1 - 46.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.1T + 1.21e4T^{2} \) |
| 31 | \( 1 + 47.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 480.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 551.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 628.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 693.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 785.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 807.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 333.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 34.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 929.T + 9.12e5T^{2} \) |
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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
−9.059816560382703990189529311097, −8.396582409098441278128088360963, −6.87163838998507193173051406154, −6.35630030672496630797570182472, −5.94947782714920824620389586393, −5.01903068084942259159326350892, −3.69723457791921333521543293078, −2.97462948498979709109657966227, −1.81508929596253478815442226177, −0.73781794159865002951838995447,
0.73781794159865002951838995447, 1.81508929596253478815442226177, 2.97462948498979709109657966227, 3.69723457791921333521543293078, 5.01903068084942259159326350892, 5.94947782714920824620389586393, 6.35630030672496630797570182472, 6.87163838998507193173051406154, 8.396582409098441278128088360963, 9.059816560382703990189529311097