L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s − 2·9-s − 2·10-s + 2·12-s − 4·13-s − 4·14-s + 15-s − 4·16-s − 2·17-s + 4·18-s + 2·20-s + 2·21-s − 23-s − 4·25-s + 8·26-s − 5·27-s + 4·28-s − 2·30-s − 7·31-s + 8·32-s + 4·34-s + 2·35-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.577·12-s − 1.10·13-s − 1.06·14-s + 0.258·15-s − 16-s − 0.485·17-s + 0.942·18-s + 0.447·20-s + 0.436·21-s − 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s + 0.755·28-s − 0.365·30-s − 1.25·31-s + 1.41·32-s + 0.685·34-s + 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.30995332857195, −13.82159656657588, −13.27869330655223, −12.90713695615587, −11.92332889598878, −11.71667759787844, −11.15136563809049, −10.69926818322506, −10.06192663325153, −9.677114237131273, −9.337391717720302, −8.713115312103717, −8.304940870999106, −7.983870561076161, −7.436288419510423, −6.864351659901364, −6.418354431906099, −5.390498998873731, −5.243942080125882, −4.452643648861607, −3.757791182342200, −2.976130434446346, −2.318120435707643, −1.829646753752124, −1.432353660361264, 0, 0,
1.432353660361264, 1.829646753752124, 2.318120435707643, 2.976130434446346, 3.757791182342200, 4.452643648861607, 5.243942080125882, 5.390498998873731, 6.418354431906099, 6.864351659901364, 7.436288419510423, 7.983870561076161, 8.304940870999106, 8.713115312103717, 9.337391717720302, 9.677114237131273, 10.06192663325153, 10.69926818322506, 11.15136563809049, 11.71667759787844, 11.92332889598878, 12.90713695615587, 13.27869330655223, 13.82159656657588, 14.30995332857195