| L(s) = 1 | − 8.60·3-s + 17.7·7-s + 47.0·9-s + 29.0·11-s + 54.6·13-s + 68.3·17-s − 53.7·19-s − 152.·21-s − 166.·23-s − 172.·27-s − 255.·29-s + 206.·31-s − 249.·33-s − 243.·37-s − 470.·39-s − 47.2·41-s − 35.8·43-s + 467.·47-s − 27.8·49-s − 588.·51-s − 673.·53-s + 462.·57-s − 330.·59-s − 396.·61-s + 835.·63-s + 356.·67-s + 1.43e3·69-s + ⋯ |
| L(s) = 1 | − 1.65·3-s + 0.958·7-s + 1.74·9-s + 0.795·11-s + 1.16·13-s + 0.975·17-s − 0.649·19-s − 1.58·21-s − 1.51·23-s − 1.22·27-s − 1.63·29-s + 1.19·31-s − 1.31·33-s − 1.08·37-s − 1.92·39-s − 0.180·41-s − 0.127·43-s + 1.45·47-s − 0.0811·49-s − 1.61·51-s − 1.74·53-s + 1.07·57-s − 0.729·59-s − 0.831·61-s + 1.67·63-s + 0.649·67-s + 2.50·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8.60T + 27T^{2} \) |
| 7 | \( 1 - 17.7T + 343T^{2} \) |
| 11 | \( 1 - 29.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 255.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 330.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 396.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 356.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 337.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 727.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 64.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 698.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 964.T + 9.12e5T^{2} \) |
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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
−8.086307503242650689012114647130, −7.34293441650458884771124117254, −6.25433830666074343421164843410, −6.01752815795767000550615308604, −5.14425166777009085413933481138, −4.34191039834050114781134677085, −3.61597269597653394033638662183, −1.78770172646844356172793275034, −1.17276364868100865122444991484, 0,
1.17276364868100865122444991484, 1.78770172646844356172793275034, 3.61597269597653394033638662183, 4.34191039834050114781134677085, 5.14425166777009085413933481138, 6.01752815795767000550615308604, 6.25433830666074343421164843410, 7.34293441650458884771124117254, 8.086307503242650689012114647130
