| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s − 2·11-s + 2·12-s − 6·13-s − 4·16-s + 17-s + 2·18-s − 2·19-s − 4·22-s + 23-s − 12·26-s + 27-s − 29-s + 5·31-s − 8·32-s − 2·33-s + 2·34-s + 2·36-s + 7·37-s − 4·38-s − 6·39-s + 7·41-s + 8·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.66·13-s − 16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s − 0.852·22-s + 0.208·23-s − 2.35·26-s + 0.192·27-s − 0.185·29-s + 0.898·31-s − 1.41·32-s − 0.348·33-s + 0.342·34-s + 1/3·36-s + 1.15·37-s − 0.648·38-s − 0.960·39-s + 1.09·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.042319228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.042319228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.00522831772509, −13.42510886802801, −12.90818142420680, −12.66588317793001, −12.20918088430201, −11.65125975710806, −11.10688844783691, −10.54643276860325, −9.893150322966935, −9.448857578475337, −9.041088907413492, −8.194445374980471, −7.687231184786016, −7.391820737347910, −6.526099465792665, −6.228905896568310, −5.465297829765644, −4.971415204941465, −4.467537054835944, −4.148986902042520, −3.200048661024054, −2.810764303858462, −2.416285666870758, −1.649663430354728, −0.4542631129577599,
0.4542631129577599, 1.649663430354728, 2.416285666870758, 2.810764303858462, 3.200048661024054, 4.148986902042520, 4.467537054835944, 4.971415204941465, 5.465297829765644, 6.228905896568310, 6.526099465792665, 7.391820737347910, 7.687231184786016, 8.194445374980471, 9.041088907413492, 9.448857578475337, 9.893150322966935, 10.54643276860325, 11.10688844783691, 11.65125975710806, 12.20918088430201, 12.66588317793001, 12.90818142420680, 13.42510886802801, 14.00522831772509
