| L(s) = 1 | + 3.29·3-s − 1.54·7-s + 7.83·9-s + 3.54·11-s − 2.25·13-s + 17-s − 4.83·19-s − 5.09·21-s + 1.54·23-s + 15.9·27-s − 2.83·29-s + 7.87·31-s + 11.6·33-s + 0.584·37-s − 7.42·39-s + 9.09·41-s + 2·43-s − 6.83·47-s − 4.60·49-s + 3.29·51-s + 12.9·53-s − 15.9·57-s + 13.3·59-s + 3.74·61-s − 12.1·63-s − 10.2·67-s + 5.09·69-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.584·7-s + 2.61·9-s + 1.06·11-s − 0.625·13-s + 0.242·17-s − 1.11·19-s − 1.11·21-s + 0.322·23-s + 3.06·27-s − 0.527·29-s + 1.41·31-s + 2.03·33-s + 0.0961·37-s − 1.18·39-s + 1.42·41-s + 0.304·43-s − 0.997·47-s − 0.657·49-s + 0.461·51-s + 1.77·53-s − 2.11·57-s + 1.73·59-s + 0.479·61-s − 1.52·63-s − 1.25·67-s + 0.613·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.442276905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.442276905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 - 3.54T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 - 7.87T + 31T^{2} \) |
| 37 | \( 1 - 0.584T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4.78T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.138403027316202347225957881814, −7.33224181052779567092260208887, −6.80756358230150049994649966868, −6.11453255021216040276293556460, −4.80722010911119252737566414249, −4.07431551191088779423518125562, −3.57157677493960960492138982087, −2.68056045283070700104839067405, −2.13521988275044126963617806068, −1.02521189434480575516928809153,
1.02521189434480575516928809153, 2.13521988275044126963617806068, 2.68056045283070700104839067405, 3.57157677493960960492138982087, 4.07431551191088779423518125562, 4.80722010911119252737566414249, 6.11453255021216040276293556460, 6.80756358230150049994649966868, 7.33224181052779567092260208887, 8.138403027316202347225957881814
