L(s) = 1 | − 3-s − 4.42·5-s − 0.622·7-s + 9-s − 5.80·11-s + 2·13-s + 4.42·15-s + 1.37·17-s − 6.42·19-s + 0.622·21-s + 23-s + 14.6·25-s − 27-s − 0.755·29-s + 1.24·31-s + 5.80·33-s + 2.75·35-s + 9.05·37-s − 2·39-s − 9.61·41-s + 5.18·43-s − 4.42·45-s + 5.24·47-s − 6.61·49-s − 1.37·51-s + 12.0·53-s + 25.7·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.98·5-s − 0.235·7-s + 0.333·9-s − 1.75·11-s + 0.554·13-s + 1.14·15-s + 0.334·17-s − 1.47·19-s + 0.135·21-s + 0.208·23-s + 2.92·25-s − 0.192·27-s − 0.140·29-s + 0.223·31-s + 1.01·33-s + 0.465·35-s + 1.48·37-s − 0.320·39-s − 1.50·41-s + 0.790·43-s − 0.660·45-s + 0.764·47-s − 0.944·49-s − 0.192·51-s + 1.65·53-s + 3.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5126783215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5126783215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4.42T + 5T^{2} \) |
| 7 | \( 1 + 0.622T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−10.19928258656506714078037117707, −8.718651993633261235742664101418, −8.120686891753311086327396512929, −7.46955464318567561692288637246, −6.62623392310406154252545215878, −5.47955888416803786877964722493, −4.54397226023140078323647354917, −3.79086309884720451163885921534, −2.68544767092925493409204352717, −0.53551138840216696430331899125,
0.53551138840216696430331899125, 2.68544767092925493409204352717, 3.79086309884720451163885921534, 4.54397226023140078323647354917, 5.47955888416803786877964722493, 6.62623392310406154252545215878, 7.46955464318567561692288637246, 8.120686891753311086327396512929, 8.718651993633261235742664101418, 10.19928258656506714078037117707