| L(s) = 1 | − 3.45·5-s − 7·7-s − 27.2·11-s + 58.7·13-s − 49.2·17-s + 157.·19-s − 82.1·23-s − 113.·25-s + 194.·29-s − 115.·31-s + 24.1·35-s + 327.·37-s + 136.·41-s − 311.·43-s − 355.·47-s + 49·49-s − 677.·53-s + 94.1·55-s − 197.·59-s + 61.0·61-s − 202.·65-s − 1.01e3·67-s − 279.·71-s + 629.·73-s + 191.·77-s − 20.2·79-s − 260.·83-s + ⋯ |
| L(s) = 1 | − 0.308·5-s − 0.377·7-s − 0.748·11-s + 1.25·13-s − 0.703·17-s + 1.89·19-s − 0.745·23-s − 0.904·25-s + 1.24·29-s − 0.670·31-s + 0.116·35-s + 1.45·37-s + 0.518·41-s − 1.10·43-s − 1.10·47-s + 0.142·49-s − 1.75·53-s + 0.230·55-s − 0.435·59-s + 0.128·61-s − 0.386·65-s − 1.85·67-s − 0.467·71-s + 1.00·73-s + 0.282·77-s − 0.0287·79-s − 0.344·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 + 3.45T + 125T^{2} \) |
| 11 | \( 1 + 27.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 157.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 355.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 677.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 197.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 61.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 20.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 260.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 909.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 100.T + 9.12e5T^{2} \) |
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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
−9.262623395784586626866541296730, −8.174985530316893374518834669012, −7.68045345636541897809784214225, −6.53184197001622102522329749469, −5.80348144514888075270200924616, −4.75530245899297395353123375720, −3.68333725128820021714066333282, −2.82451486012500236500561417639, −1.36976374470415756234981889345, 0,
1.36976374470415756234981889345, 2.82451486012500236500561417639, 3.68333725128820021714066333282, 4.75530245899297395353123375720, 5.80348144514888075270200924616, 6.53184197001622102522329749469, 7.68045345636541897809784214225, 8.174985530316893374518834669012, 9.262623395784586626866541296730
