| L(s) = 1 | − 2.43·3-s + 9.42·5-s + 9.58·7-s − 21.0·9-s − 37.8·11-s + 30.7·13-s − 22.9·15-s + 35.3·17-s − 19·19-s − 23.3·21-s + 88.7·23-s − 36.1·25-s + 117.·27-s + 226.·29-s − 320.·31-s + 92.2·33-s + 90.3·35-s − 346.·37-s − 75.0·39-s − 150.·41-s − 284.·43-s − 198.·45-s − 240.·47-s − 251.·49-s − 86.1·51-s + 539.·53-s − 356.·55-s + ⋯ |
| L(s) = 1 | − 0.469·3-s + 0.843·5-s + 0.517·7-s − 0.779·9-s − 1.03·11-s + 0.656·13-s − 0.395·15-s + 0.504·17-s − 0.229·19-s − 0.242·21-s + 0.804·23-s − 0.289·25-s + 0.835·27-s + 1.44·29-s − 1.85·31-s + 0.486·33-s + 0.436·35-s − 1.54·37-s − 0.308·39-s − 0.574·41-s − 1.00·43-s − 0.657·45-s − 0.746·47-s − 0.731·49-s − 0.236·51-s + 1.39·53-s − 0.875·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 2.43T + 27T^{2} \) |
| 5 | \( 1 - 9.42T + 125T^{2} \) |
| 7 | \( 1 - 9.58T + 343T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.3T + 4.91e3T^{2} \) |
| 23 | \( 1 - 88.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 320.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 539.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 377.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 78.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 924.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 601.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 828.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 101.T + 9.12e5T^{2} \) |
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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
−8.745697387393247532207518517755, −8.369347225944152191615515585359, −7.20909239354645916163485047178, −6.30993358739339827970460142286, −5.34172705130033227256274871946, −5.15072503554579312871532524176, −3.56630624600587041116109060431, −2.50108132990059078587805554910, −1.40795347655610469102204979296, 0,
1.40795347655610469102204979296, 2.50108132990059078587805554910, 3.56630624600587041116109060431, 5.15072503554579312871532524176, 5.34172705130033227256274871946, 6.30993358739339827970460142286, 7.20909239354645916163485047178, 8.369347225944152191615515585359, 8.745697387393247532207518517755
