| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 5·11-s + 5·13-s + 14-s + 16-s + 8·17-s − 8·19-s − 5·22-s + 23-s + 5·26-s + 28-s + 2·29-s + 32-s + 8·34-s − 3·37-s − 8·38-s − 6·41-s + 4·43-s − 5·44-s + 46-s + 9·47-s + 49-s + 5·52-s + 8·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.50·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s − 1.83·19-s − 1.06·22-s + 0.208·23-s + 0.980·26-s + 0.188·28-s + 0.371·29-s + 0.176·32-s + 1.37·34-s − 0.493·37-s − 1.29·38-s − 0.937·41-s + 0.609·43-s − 0.753·44-s + 0.147·46-s + 1.31·47-s + 1/7·49-s + 0.693·52-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.574705605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.574705605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.69378755230134465018163687351, −6.97598181652209453665565436349, −6.12443850174224758472986079194, −5.58555985499331257974970754483, −5.05949288536274815753632926108, −4.15247860442060912091569923905, −3.52692174462644235053179589826, −2.71106472574327609603536566549, −1.89734095189854141358220435128, −0.820907458239425984291945204244,
0.820907458239425984291945204244, 1.89734095189854141358220435128, 2.71106472574327609603536566549, 3.52692174462644235053179589826, 4.15247860442060912091569923905, 5.05949288536274815753632926108, 5.58555985499331257974970754483, 6.12443850174224758472986079194, 6.97598181652209453665565436349, 7.69378755230134465018163687351
