| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 3·7-s + 9-s − 2·10-s + 11-s + 2·12-s − 2·13-s − 6·14-s − 15-s − 4·16-s + 2·17-s + 2·18-s + 6·19-s − 2·20-s − 3·21-s + 2·22-s + 3·23-s − 4·25-s − 4·26-s + 27-s − 6·28-s + 3·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s − 1.60·14-s − 0.258·15-s − 16-s + 0.485·17-s + 0.471·18-s + 1.37·19-s − 0.447·20-s − 0.654·21-s + 0.426·22-s + 0.625·23-s − 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.13·28-s + 0.557·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107535578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107535578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.27824273167304406003722016011, −12.33792187853331891937545023201, −11.68018802766311299547378241988, −10.03012465082032508207642749183, −9.085897389420589721942660476599, −7.52539857035144694302700552711, −6.46487870715497229983731599323, −5.14885571821443324588162204875, −3.77257881491072191084851286374, −2.93657054326743540839942565323,
2.93657054326743540839942565323, 3.77257881491072191084851286374, 5.14885571821443324588162204875, 6.46487870715497229983731599323, 7.52539857035144694302700552711, 9.085897389420589721942660476599, 10.03012465082032508207642749183, 11.68018802766311299547378241988, 12.33792187853331891937545023201, 13.27824273167304406003722016011
