| L(s) = 1 | + 4·3-s − 4·7-s − 11·9-s + 43.0·11-s − 21.5·13-s − 43.0·17-s − 129.·19-s − 16·21-s + 52·23-s − 152·27-s − 158·29-s + 172.·31-s + 172.·33-s + 280.·37-s − 86.1·39-s − 170·41-s − 316·43-s − 244·47-s − 327·49-s − 172.·51-s − 495.·53-s − 516.·57-s + 646.·59-s + 82·61-s + 44·63-s − 692·67-s + 208·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.215·7-s − 0.407·9-s + 1.18·11-s − 0.459·13-s − 0.614·17-s − 1.56·19-s − 0.166·21-s + 0.471·23-s − 1.08·27-s − 1.01·29-s + 0.998·31-s + 0.909·33-s + 1.24·37-s − 0.353·39-s − 0.647·41-s − 1.12·43-s − 0.757·47-s − 0.953·49-s − 0.473·51-s − 1.28·53-s − 1.20·57-s + 1.42·59-s + 0.172·61-s + 0.0879·63-s − 1.26·67-s + 0.362·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4T + 27T^{2} \) |
| 7 | \( 1 + 4T + 343T^{2} \) |
| 11 | \( 1 - 43.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52T + 1.21e4T^{2} \) |
| 29 | \( 1 + 158T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 170T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316T + 7.95e4T^{2} \) |
| 47 | \( 1 + 244T + 1.03e5T^{2} \) |
| 53 | \( 1 + 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 646.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82T + 2.26e5T^{2} \) |
| 67 | \( 1 + 692T + 3.00e5T^{2} \) |
| 71 | \( 1 + 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 344.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 940T + 5.71e5T^{2} \) |
| 89 | \( 1 - 6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−9.287240871899729315641205513772, −8.685937561397672258895861497817, −7.926260230411318492674464067068, −6.76414057158544763348522265944, −6.14639696963288802742516766422, −4.75619992829946513282469501778, −3.82351669212760419386627883124, −2.79526411175839626968638715703, −1.73716366806993572092927253180, 0,
1.73716366806993572092927253180, 2.79526411175839626968638715703, 3.82351669212760419386627883124, 4.75619992829946513282469501778, 6.14639696963288802742516766422, 6.76414057158544763348522265944, 7.926260230411318492674464067068, 8.685937561397672258895861497817, 9.287240871899729315641205513772
