| L(s) = 1 | + 2.69·3-s + 4.28·9-s + 0.919·11-s − 5.24·13-s − 4.25·17-s + 1.83·19-s + 8.78·23-s + 3.46·27-s + 6.71·29-s − 4.64·31-s + 2.48·33-s + 1.42·37-s − 14.1·39-s − 7.35·41-s + 9.80·43-s + 6.80·47-s − 11.4·51-s + 11.2·53-s + 4.96·57-s + 11.3·59-s − 9.64·61-s + 2.78·67-s + 23.6·69-s + 11.8·71-s − 5.11·73-s + 0.727·79-s − 3.50·81-s + ⋯ |
| L(s) = 1 | + 1.55·3-s + 1.42·9-s + 0.277·11-s − 1.45·13-s − 1.03·17-s + 0.422·19-s + 1.83·23-s + 0.666·27-s + 1.24·29-s − 0.833·31-s + 0.432·33-s + 0.233·37-s − 2.26·39-s − 1.14·41-s + 1.49·43-s + 0.992·47-s − 1.60·51-s + 1.54·53-s + 0.657·57-s + 1.47·59-s − 1.23·61-s + 0.340·67-s + 2.85·69-s + 1.40·71-s − 0.598·73-s + 0.0818·79-s − 0.389·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.774534446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.774534446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.69T + 3T^{2} \) |
| 11 | \( 1 - 0.919T + 11T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 8.78T + 23T^{2} \) |
| 29 | \( 1 - 6.71T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 - 2.78T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 0.727T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 + 0.963T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.58806910183627071595754702567, −7.18587379389307918106375029275, −6.66081117840596215495004573222, −5.46779542578229136146541176109, −4.75513172630937746001450251071, −4.09982726872950057973298729141, −3.24046785189643203721833274155, −2.59232381128793649155622828634, −2.07962222167241745579182379487, −0.847544834865845520376875412355,
0.847544834865845520376875412355, 2.07962222167241745579182379487, 2.59232381128793649155622828634, 3.24046785189643203721833274155, 4.09982726872950057973298729141, 4.75513172630937746001450251071, 5.46779542578229136146541176109, 6.66081117840596215495004573222, 7.18587379389307918106375029275, 7.58806910183627071595754702567
