| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 4·7-s − 3·8-s + 9-s − 12-s − 6·13-s − 4·14-s − 16-s + 6·17-s + 18-s + 2·19-s − 4·21-s − 4·23-s − 3·24-s − 6·26-s + 27-s + 4·28-s + 8·29-s + 6·31-s + 5·32-s + 6·34-s − 36-s + 6·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s − 0.834·23-s − 0.612·24-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + 1.48·29-s + 1.07·31-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.986·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.852212391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.852212391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.543359650662236098417331516923, −7.86786008362572605457503887932, −6.99294957245247344706144578433, −6.26711155117347263403320630315, −5.46249218413326480091048866367, −4.68802659153404688563028812078, −3.83440269089398061389622899594, −3.06046767146419013698609448578, −2.57098947113755432400667040759, −0.67892565669524491181453663738,
0.67892565669524491181453663738, 2.57098947113755432400667040759, 3.06046767146419013698609448578, 3.83440269089398061389622899594, 4.68802659153404688563028812078, 5.46249218413326480091048866367, 6.26711155117347263403320630315, 6.99294957245247344706144578433, 7.86786008362572605457503887932, 8.543359650662236098417331516923
