| L(s) = 1 | − 3·2-s + 4-s − 5·5-s + 21·8-s + 15·10-s + 24·11-s − 74·13-s − 71·16-s + 54·17-s + 124·19-s − 5·20-s − 72·22-s + 120·23-s + 25·25-s + 222·26-s + 78·29-s − 200·31-s + 45·32-s − 162·34-s − 70·37-s − 372·38-s − 105·40-s + 330·41-s + 92·43-s + 24·44-s − 360·46-s − 24·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.447·5-s + 0.928·8-s + 0.474·10-s + 0.657·11-s − 1.57·13-s − 1.10·16-s + 0.770·17-s + 1.49·19-s − 0.0559·20-s − 0.697·22-s + 1.08·23-s + 1/5·25-s + 1.67·26-s + 0.499·29-s − 1.15·31-s + 0.248·32-s − 0.817·34-s − 0.311·37-s − 1.58·38-s − 0.415·40-s + 1.25·41-s + 0.326·43-s + 0.0822·44-s − 1.15·46-s − 0.0744·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9379994835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9379994835\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 24 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−8.872183857462422900299313046457, −7.79316687641168669154280175365, −7.48282949202352947789234776692, −6.77096220357683672215251691479, −5.38744858113774301145412329362, −4.78717404912861731691706979711, −3.74705321285733907124172301911, −2.71506517231415179182608794675, −1.41146798293449679129928343468, −0.56600631635234203978422130342,
0.56600631635234203978422130342, 1.41146798293449679129928343468, 2.71506517231415179182608794675, 3.74705321285733907124172301911, 4.78717404912861731691706979711, 5.38744858113774301145412329362, 6.77096220357683672215251691479, 7.48282949202352947789234776692, 7.79316687641168669154280175365, 8.872183857462422900299313046457
