L(s) = 1 | − 0.532·3-s − 2.53·5-s + 1.87·7-s − 2.71·9-s − 3.41·11-s − 5.29·13-s + 1.34·15-s + 1.65·17-s − 21-s − 1.75·23-s + 1.41·25-s + 3.04·27-s − 3.46·29-s − 1.94·31-s + 1.81·33-s − 4.75·35-s − 0.837·37-s + 2.81·39-s − 4.49·41-s − 4.80·43-s + 6.87·45-s − 0.716·47-s − 3.46·49-s − 0.879·51-s − 6.10·53-s + 8.63·55-s + 10.7·59-s + ⋯ |
L(s) = 1 | − 0.307·3-s − 1.13·5-s + 0.710·7-s − 0.905·9-s − 1.02·11-s − 1.46·13-s + 0.347·15-s + 0.400·17-s − 0.218·21-s − 0.366·23-s + 0.282·25-s + 0.585·27-s − 0.643·29-s − 0.349·31-s + 0.315·33-s − 0.804·35-s − 0.137·37-s + 0.450·39-s − 0.701·41-s − 0.732·43-s + 1.02·45-s − 0.104·47-s − 0.495·49-s − 0.123·51-s − 0.838·53-s + 1.16·55-s + 1.40·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4380758016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4380758016\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 0.532T + 3T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 + 0.837T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 + 0.716T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 - 6.96T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
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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
−8.051107583024498827481833711096, −7.55477673113982905445223853756, −6.89157783028650655086123135443, −5.77321073049540051199006914241, −5.13615831325353909528560780176, −4.66684637280649323537902558173, −3.64081001879820779915753759079, −2.84951476002161544458758095909, −1.93927880283720864446722102160, −0.33562168060467583384510505309,
0.33562168060467583384510505309, 1.93927880283720864446722102160, 2.84951476002161544458758095909, 3.64081001879820779915753759079, 4.66684637280649323537902558173, 5.13615831325353909528560780176, 5.77321073049540051199006914241, 6.89157783028650655086123135443, 7.55477673113982905445223853756, 8.051107583024498827481833711096