L(s) = 1 | + 1.71·2-s − 3-s + 0.930·4-s − 2.44·5-s − 1.71·6-s − 7-s − 1.83·8-s + 9-s − 4.19·10-s − 4.01·11-s − 0.930·12-s + 2.21·13-s − 1.71·14-s + 2.44·15-s − 4.99·16-s + 5.53·17-s + 1.71·18-s − 8.54·19-s − 2.27·20-s + 21-s − 6.88·22-s − 2.62·23-s + 1.83·24-s + 0.999·25-s + 3.79·26-s − 27-s − 0.930·28-s + ⋯ |
L(s) = 1 | + 1.21·2-s − 0.577·3-s + 0.465·4-s − 1.09·5-s − 0.698·6-s − 0.377·7-s − 0.647·8-s + 0.333·9-s − 1.32·10-s − 1.21·11-s − 0.268·12-s + 0.614·13-s − 0.457·14-s + 0.632·15-s − 1.24·16-s + 1.34·17-s + 0.403·18-s − 1.95·19-s − 0.509·20-s + 0.218·21-s − 1.46·22-s − 0.547·23-s + 0.373·24-s + 0.199·25-s + 0.743·26-s − 0.192·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7692017911\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7692017911\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 8.54T + 19T^{2} \) |
| 23 | \( 1 + 2.62T + 23T^{2} \) |
| 29 | \( 1 - 1.50T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 - 8.35T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 + 8.47T + 83T^{2} \) |
| 89 | \( 1 + 0.821T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.67511731876284544778387286254, −7.02653420615263643222791111420, −6.10695330353990321818528302200, −5.73706433076287092806099254313, −4.97257692043099867543763579014, −4.25993443924831201546511061776, −3.69887826605911235065205622491, −3.08829638163085232332605226242, −2.00189841101825322871355968811, −0.35010349639405530809068291390,
0.35010349639405530809068291390, 2.00189841101825322871355968811, 3.08829638163085232332605226242, 3.69887826605911235065205622491, 4.25993443924831201546511061776, 4.97257692043099867543763579014, 5.73706433076287092806099254313, 6.10695330353990321818528302200, 7.02653420615263643222791111420, 7.67511731876284544778387286254