| L(s) = 1 | − 1.66·2-s − 1.56·3-s + 0.760·4-s − 0.618·5-s + 2.60·6-s + 1.60·7-s + 2.05·8-s − 0.540·9-s + 1.02·10-s + 11-s − 1.19·12-s − 4.28·13-s − 2.67·14-s + 0.970·15-s − 4.94·16-s − 2.36·17-s + 0.898·18-s − 7.82·19-s − 0.470·20-s − 2.52·21-s − 1.66·22-s − 3.22·24-s − 4.61·25-s + 7.11·26-s + 5.55·27-s + 1.22·28-s − 9.55·29-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 0.905·3-s + 0.380·4-s − 0.276·5-s + 1.06·6-s + 0.608·7-s + 0.728·8-s − 0.180·9-s + 0.325·10-s + 0.301·11-s − 0.344·12-s − 1.18·13-s − 0.714·14-s + 0.250·15-s − 1.23·16-s − 0.574·17-s + 0.211·18-s − 1.79·19-s − 0.105·20-s − 0.550·21-s − 0.354·22-s − 0.659·24-s − 0.923·25-s + 1.39·26-s + 1.06·27-s + 0.231·28-s − 1.77·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05308435163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05308435163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 + 7.82T + 19T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 + 8.68T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 4.99T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 0.722T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.06T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 3.97T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−8.179617850476327272918017306513, −7.43073353129963776086029974008, −6.97010058956489748366377195003, −6.01961991339931643217387370611, −5.27801404582693633870599398133, −4.53730556330152525270862469018, −3.87981181461340546740423077023, −2.30858745797831113361236743584, −1.66137668027076078250751606765, −0.15010173277645777575251467316,
0.15010173277645777575251467316, 1.66137668027076078250751606765, 2.30858745797831113361236743584, 3.87981181461340546740423077023, 4.53730556330152525270862469018, 5.27801404582693633870599398133, 6.01961991339931643217387370611, 6.97010058956489748366377195003, 7.43073353129963776086029974008, 8.179617850476327272918017306513
