L(s) = 1 | + 2.15·2-s + 2.62·3-s + 2.64·4-s + 5.65·6-s + 7-s + 1.39·8-s + 3.88·9-s + 3.15·11-s + 6.94·12-s + 1.72·13-s + 2.15·14-s − 2.28·16-s + 5.38·17-s + 8.37·18-s − 3.42·19-s + 2.62·21-s + 6.80·22-s − 23-s + 3.65·24-s + 3.72·26-s + 2.32·27-s + 2.64·28-s − 2.54·29-s − 9.96·31-s − 7.72·32-s + 8.28·33-s + 11.6·34-s + ⋯ |
L(s) = 1 | + 1.52·2-s + 1.51·3-s + 1.32·4-s + 2.30·6-s + 0.377·7-s + 0.492·8-s + 1.29·9-s + 0.951·11-s + 2.00·12-s + 0.479·13-s + 0.576·14-s − 0.572·16-s + 1.30·17-s + 1.97·18-s − 0.786·19-s + 0.572·21-s + 1.45·22-s − 0.208·23-s + 0.746·24-s + 0.731·26-s + 0.446·27-s + 0.500·28-s − 0.471·29-s − 1.79·31-s − 1.36·32-s + 1.44·33-s + 1.99·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.506944478\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.506944478\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 - 7.30T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 - 3.20T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 + 5.20T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 4.05T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 + 4.36T + 97T^{2} \) |
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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
−8.412270440098431439194727290302, −7.61942149001274668470954684709, −6.98974067022530953276596287167, −6.00266084422065133348796751439, −5.44024190877397346366028629487, −4.22701204152722738470172863806, −3.91420671998509982133218308342, −3.20798793580152778132755992475, −2.35772052916847044614722097571, −1.49149066884146795578000399191,
1.49149066884146795578000399191, 2.35772052916847044614722097571, 3.20798793580152778132755992475, 3.91420671998509982133218308342, 4.22701204152722738470172863806, 5.44024190877397346366028629487, 6.00266084422065133348796751439, 6.98974067022530953276596287167, 7.61942149001274668470954684709, 8.412270440098431439194727290302