L(s) = 1 | + 3·3-s − 2·5-s + 6·9-s + 11-s + 7·13-s − 6·15-s − 2·17-s + 8·23-s − 25-s + 9·27-s − 5·29-s + 4·31-s + 3·33-s + 4·37-s + 21·39-s − 4·41-s + 8·43-s − 12·45-s + 2·47-s − 6·51-s − 6·53-s − 2·55-s + 3·59-s − 61-s − 14·65-s − 9·67-s + 24·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.894·5-s + 2·9-s + 0.301·11-s + 1.94·13-s − 1.54·15-s − 0.485·17-s + 1.66·23-s − 1/5·25-s + 1.73·27-s − 0.928·29-s + 0.718·31-s + 0.522·33-s + 0.657·37-s + 3.36·39-s − 0.624·41-s + 1.21·43-s − 1.78·45-s + 0.291·47-s − 0.840·51-s − 0.824·53-s − 0.269·55-s + 0.390·59-s − 0.128·61-s − 1.73·65-s − 1.09·67-s + 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.132357372\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.132357372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.83650573755773280953231464549, −7.36375735223665914047822982700, −6.62174297748309329215976154025, −5.81586884475129280141501808903, −4.56403633053091797282961632728, −4.03910882013454900631081638464, −3.41930656959819350421158756845, −2.90134032691859498509709201219, −1.82930615798941828265158263410, −0.968948037648386637365537444731,
0.968948037648386637365537444731, 1.82930615798941828265158263410, 2.90134032691859498509709201219, 3.41930656959819350421158756845, 4.03910882013454900631081638464, 4.56403633053091797282961632728, 5.81586884475129280141501808903, 6.62174297748309329215976154025, 7.36375735223665914047822982700, 7.83650573755773280953231464549