| L(s) = 1 | − 3.56·2-s + 4.68·4-s − 7·7-s + 11.8·8-s + 5.19·11-s + 54.5·13-s + 24.9·14-s − 79.5·16-s − 16.1·17-s + 87.4·19-s − 18.4·22-s − 176.·23-s − 194.·26-s − 32.7·28-s − 142.·29-s − 94.3·31-s + 188.·32-s + 57.5·34-s + 17.3·37-s − 311.·38-s − 210.·41-s + 521.·43-s + 24.3·44-s + 628.·46-s + 105.·47-s + 49·49-s + 255.·52-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.585·4-s − 0.377·7-s + 0.521·8-s + 0.142·11-s + 1.16·13-s + 0.475·14-s − 1.24·16-s − 0.230·17-s + 1.05·19-s − 0.179·22-s − 1.59·23-s − 1.46·26-s − 0.221·28-s − 0.910·29-s − 0.546·31-s + 1.04·32-s + 0.290·34-s + 0.0770·37-s − 1.32·38-s − 0.800·41-s + 1.84·43-s + 0.0833·44-s + 2.01·46-s + 0.327·47-s + 0.142·49-s + 0.681·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 3.56T + 8T^{2} \) |
| 11 | \( 1 - 5.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 17.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 521.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 793.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 425.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 843.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−8.886840287689155442403376090189, −7.903381587939611248249038603958, −7.41270878747009774161035356039, −6.36762772989215390310165875756, −5.61200252592133452245053393023, −4.31355506491736231303404795455, −3.46660110052410998052768728014, −2.07257940273831589471743712301, −1.10409410526093532743115529962, 0,
1.10409410526093532743115529962, 2.07257940273831589471743712301, 3.46660110052410998052768728014, 4.31355506491736231303404795455, 5.61200252592133452245053393023, 6.36762772989215390310165875756, 7.41270878747009774161035356039, 7.903381587939611248249038603958, 8.886840287689155442403376090189
