L(s) = 1 | − 2.06·2-s + 3-s + 2.25·4-s − 1.75·5-s − 2.06·6-s − 2.34·7-s − 0.525·8-s + 9-s + 3.61·10-s − 5.74·11-s + 2.25·12-s + 5.83·13-s + 4.84·14-s − 1.75·15-s − 3.42·16-s − 2.97·17-s − 2.06·18-s − 6.07·19-s − 3.95·20-s − 2.34·21-s + 11.8·22-s − 2.78·23-s − 0.525·24-s − 1.92·25-s − 12.0·26-s + 27-s − 5.29·28-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 0.577·3-s + 1.12·4-s − 0.784·5-s − 0.842·6-s − 0.887·7-s − 0.185·8-s + 0.333·9-s + 1.14·10-s − 1.73·11-s + 0.650·12-s + 1.61·13-s + 1.29·14-s − 0.452·15-s − 0.856·16-s − 0.720·17-s − 0.486·18-s − 1.39·19-s − 0.883·20-s − 0.512·21-s + 2.52·22-s − 0.581·23-s − 0.107·24-s − 0.385·25-s − 2.36·26-s + 0.192·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1876720516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1876720516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 + 8.99T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 8.04T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.042289679644994264789767304524, −7.55075777148377202107030740335, −6.58556812486087356574280512446, −6.22255082170064862521377494939, −4.95365596925489936147247043846, −4.06574636186243668101421039110, −3.37436083597680397780602864225, −2.46707228554028643579458308879, −1.66796756769422961231172910307, −0.25141235938350988237714857799,
0.25141235938350988237714857799, 1.66796756769422961231172910307, 2.46707228554028643579458308879, 3.37436083597680397780602864225, 4.06574636186243668101421039110, 4.95365596925489936147247043846, 6.22255082170064862521377494939, 6.58556812486087356574280512446, 7.55075777148377202107030740335, 8.042289679644994264789767304524