| L(s) = 1 | − 8.40·3-s − 5.33·5-s + 23.8·7-s + 43.7·9-s − 58.6·11-s − 13·13-s + 44.8·15-s − 110.·17-s + 143.·19-s − 200.·21-s + 138.·23-s − 96.5·25-s − 140.·27-s + 256.·29-s + 14.9·31-s + 493.·33-s − 127.·35-s − 155.·37-s + 109.·39-s + 339.·41-s + 231.·43-s − 233.·45-s − 135.·47-s + 223.·49-s + 932.·51-s − 277.·53-s + 313.·55-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 0.477·5-s + 1.28·7-s + 1.61·9-s − 1.60·11-s − 0.277·13-s + 0.772·15-s − 1.58·17-s + 1.72·19-s − 2.08·21-s + 1.25·23-s − 0.772·25-s − 1.00·27-s + 1.64·29-s + 0.0864·31-s + 2.60·33-s − 0.613·35-s − 0.690·37-s + 0.448·39-s + 1.29·41-s + 0.819·43-s − 0.773·45-s − 0.420·47-s + 0.652·49-s + 2.55·51-s − 0.718·53-s + 0.767·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 8.40T + 27T^{2} \) |
| 5 | \( 1 + 5.33T + 125T^{2} \) |
| 7 | \( 1 - 23.8T + 343T^{2} \) |
| 11 | \( 1 + 58.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 14.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 135.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 277.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 358.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 269.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 261.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 986.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 772.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 472.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 527.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 458.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
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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
−9.616303677454313616620411401672, −8.334285196488189092693245848097, −7.57338417788779194362387124709, −6.82682150260362965871709154037, −5.59182918087904984546586056450, −5.00309233351124413695927581325, −4.44481145823100893580513581577, −2.65370630795507576008232920240, −1.12870238006060115733045350827, 0,
1.12870238006060115733045350827, 2.65370630795507576008232920240, 4.44481145823100893580513581577, 5.00309233351124413695927581325, 5.59182918087904984546586056450, 6.82682150260362965871709154037, 7.57338417788779194362387124709, 8.334285196488189092693245848097, 9.616303677454313616620411401672
