L(s) = 1 | − 2.77·2-s − 3.10·3-s + 5.71·4-s − 0.966·5-s + 8.62·6-s + 4.66·7-s − 10.3·8-s + 6.65·9-s + 2.68·10-s + 0.00376·11-s − 17.7·12-s + 6.25·13-s − 12.9·14-s + 3.00·15-s + 17.1·16-s + 4.19·17-s − 18.4·18-s + 2.54·19-s − 5.51·20-s − 14.4·21-s − 0.0104·22-s + 23-s + 32.0·24-s − 4.06·25-s − 17.3·26-s − 11.3·27-s + 26.6·28-s + ⋯ |
L(s) = 1 | − 1.96·2-s − 1.79·3-s + 2.85·4-s − 0.432·5-s + 3.52·6-s + 1.76·7-s − 3.64·8-s + 2.21·9-s + 0.848·10-s + 0.00113·11-s − 5.12·12-s + 1.73·13-s − 3.46·14-s + 0.775·15-s + 4.29·16-s + 1.01·17-s − 4.35·18-s + 0.583·19-s − 1.23·20-s − 3.16·21-s − 0.00223·22-s + 0.208·23-s + 6.53·24-s − 0.813·25-s − 3.40·26-s − 2.18·27-s + 5.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6576743213\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6576743213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 - T \) |
| 349 | \( 1 + T \) |
good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 0.966T + 5T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 - 0.00376T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 + 5.09T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 9.51T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 - 0.604T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.975494311124616592279136681400, −7.30782651094144514149431224848, −6.64379860831860522145185346660, −5.80909687605155866039546758033, −5.52241479432409732539926221896, −4.44615640782299849608689173031, −3.41388336642488988011954171086, −1.89252921063646597210185742230, −1.21139041831845698892836671433, −0.74318452706670643587160037290,
0.74318452706670643587160037290, 1.21139041831845698892836671433, 1.89252921063646597210185742230, 3.41388336642488988011954171086, 4.44615640782299849608689173031, 5.52241479432409732539926221896, 5.80909687605155866039546758033, 6.64379860831860522145185346660, 7.30782651094144514149431224848, 7.975494311124616592279136681400