L(s) = 1 | + 2.63·2-s + 2.79·3-s + 4.96·4-s + 5-s + 7.36·6-s + 1.28·7-s + 7.81·8-s + 4.79·9-s + 2.63·10-s − 11-s + 13.8·12-s + 3.38·14-s + 2.79·15-s + 10.6·16-s − 5.83·17-s + 12.6·18-s + 3.59·19-s + 4.96·20-s + 3.57·21-s − 2.63·22-s + 0.492·23-s + 21.8·24-s + 25-s + 4.99·27-s + 6.35·28-s + 5.61·29-s + 7.36·30-s + ⋯ |
L(s) = 1 | + 1.86·2-s + 1.61·3-s + 2.48·4-s + 0.447·5-s + 3.00·6-s + 0.484·7-s + 2.76·8-s + 1.59·9-s + 0.834·10-s − 0.301·11-s + 3.99·12-s + 0.903·14-s + 0.720·15-s + 2.67·16-s − 1.41·17-s + 2.97·18-s + 0.824·19-s + 1.10·20-s + 0.780·21-s − 0.562·22-s + 0.102·23-s + 4.45·24-s + 0.200·25-s + 0.962·27-s + 1.20·28-s + 1.04·29-s + 1.34·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(14.97901610\) |
\(L(\frac12)\) |
\(\approx\) |
\(14.97901610\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 - 0.492T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 - 3.38T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 0.795T + 47T^{2} \) |
| 53 | \( 1 + 3.13T + 53T^{2} \) |
| 59 | \( 1 + 6.74T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.06T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 8.08T + 97T^{2} \) |
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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
−7.51820878819173205273617432794, −6.91860325759756679136809522950, −6.34827532477279355840704794019, −5.37281043575314096054954709203, −4.78504813762509771107433740896, −4.21671353114251281826647902662, −3.31892460830637403770656781616, −2.92856799266637015424952048660, −2.09669816386961341332216996764, −1.61587016178989362969609961093,
1.61587016178989362969609961093, 2.09669816386961341332216996764, 2.92856799266637015424952048660, 3.31892460830637403770656781616, 4.21671353114251281826647902662, 4.78504813762509771107433740896, 5.37281043575314096054954709203, 6.34827532477279355840704794019, 6.91860325759756679136809522950, 7.51820878819173205273617432794