| L(s) = 1 | + 3.05·3-s + 3.05·7-s + 6.31·9-s − 0.259·11-s − 2.79·13-s − 17-s + 6.10·19-s + 9.31·21-s − 7.57·23-s + 10.1·27-s − 2.51·29-s + 10.3·31-s − 0.791·33-s + 8.10·37-s − 8.51·39-s − 0.102·41-s + 1.20·43-s − 3.31·47-s + 2.31·49-s − 3.05·51-s − 12.6·53-s + 18.6·57-s − 3.48·59-s − 14.2·61-s + 19.2·63-s + 4.68·67-s − 23.1·69-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 1.15·7-s + 2.10·9-s − 0.0782·11-s − 0.774·13-s − 0.242·17-s + 1.40·19-s + 2.03·21-s − 1.57·23-s + 1.94·27-s − 0.467·29-s + 1.86·31-s − 0.137·33-s + 1.33·37-s − 1.36·39-s − 0.0160·41-s + 0.184·43-s − 0.482·47-s + 0.330·49-s − 0.427·51-s − 1.73·53-s + 2.46·57-s − 0.453·59-s − 1.81·61-s + 2.42·63-s + 0.572·67-s − 2.78·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.652031300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.652031300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.05T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 0.259T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 + 0.102T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−9.446887877326974403596151877222, −8.323101122031535135992689778338, −7.87964397616345244459986508393, −7.47057285186789067553968725577, −6.22806516606467278883574168980, −4.89846217389089886790068370043, −4.29691507750850660810932108612, −3.19332314569623243538034289368, −2.37221463321489928342622351202, −1.45457895523075657036915804987,
1.45457895523075657036915804987, 2.37221463321489928342622351202, 3.19332314569623243538034289368, 4.29691507750850660810932108612, 4.89846217389089886790068370043, 6.22806516606467278883574168980, 7.47057285186789067553968725577, 7.87964397616345244459986508393, 8.323101122031535135992689778338, 9.446887877326974403596151877222
