| L(s) = 1 | − 5·2-s + 17·4-s − 5·5-s + 16·7-s − 45·8-s + 25·10-s + 44·11-s + 78·13-s − 80·14-s + 89·16-s − 18·17-s − 28·19-s − 85·20-s − 220·22-s − 184·23-s + 25·25-s − 390·26-s + 272·28-s − 29·29-s − 224·31-s − 85·32-s + 90·34-s − 80·35-s + 254·37-s + 140·38-s + 225·40-s + 78·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 0.447·5-s + 0.863·7-s − 1.98·8-s + 0.790·10-s + 1.20·11-s + 1.66·13-s − 1.52·14-s + 1.39·16-s − 0.256·17-s − 0.338·19-s − 0.950·20-s − 2.13·22-s − 1.66·23-s + 1/5·25-s − 2.94·26-s + 1.83·28-s − 0.185·29-s − 1.29·31-s − 0.469·32-s + 0.453·34-s − 0.386·35-s + 1.12·37-s + 0.597·38-s + 0.889·40-s + 0.297·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 29 | \( 1 + p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 78 T + p^{3} T^{2} \) |
| 43 | \( 1 + 260 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 + 574 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 340 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 394 T + p^{3} T^{2} \) |
| 79 | \( 1 + 960 T + p^{3} T^{2} \) |
| 83 | \( 1 - 908 T + p^{3} T^{2} \) |
| 89 | \( 1 - 990 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1234 T + p^{3} T^{2} \) |
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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
−8.863797840009503943943455283502, −8.131183534365953856200451292880, −7.69924303482950387301903155407, −6.54800081437755376337741164995, −6.04983983804566913028569760101, −4.41558186204933310566281906471, −3.46804834126453856833586438062, −1.85050503471516195393091753506, −1.31185926918867997148460769775, 0,
1.31185926918867997148460769775, 1.85050503471516195393091753506, 3.46804834126453856833586438062, 4.41558186204933310566281906471, 6.04983983804566913028569760101, 6.54800081437755376337741164995, 7.69924303482950387301903155407, 8.131183534365953856200451292880, 8.863797840009503943943455283502
