| L(s) = 1 | + 3-s + 5-s + 2.60·7-s + 9-s − 11-s + 2.60·13-s + 15-s − 0.605·17-s − 7.21·19-s + 2.60·21-s + 25-s + 27-s + 8·29-s + 5.21·31-s − 33-s + 2.60·35-s + 11.2·37-s + 2.60·39-s + 8·41-s + 10.6·43-s + 45-s − 9.21·47-s − 0.211·49-s − 0.605·51-s + 2·53-s − 55-s − 7.21·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.984·7-s + 0.333·9-s − 0.301·11-s + 0.722·13-s + 0.258·15-s − 0.146·17-s − 1.65·19-s + 0.568·21-s + 0.200·25-s + 0.192·27-s + 1.48·29-s + 0.935·31-s − 0.174·33-s + 0.440·35-s + 1.84·37-s + 0.417·39-s + 1.24·41-s + 1.61·43-s + 0.149·45-s − 1.34·47-s − 0.0301·49-s − 0.0847·51-s + 0.274·53-s − 0.134·55-s − 0.955·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.872660861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872660861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 + 0.605T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6.60T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 - 3.39T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.732462468163192598020273793109, −8.181778850433677918944686002009, −7.58820864190194669731088043629, −6.40997546153718178503646187094, −5.96492485714835452313233742623, −4.60774043885907971676049514492, −4.35528793109392519791079201569, −2.93979649941588685509397875640, −2.18183591034351161673892935637, −1.11905910247211498743823245817,
1.11905910247211498743823245817, 2.18183591034351161673892935637, 2.93979649941588685509397875640, 4.35528793109392519791079201569, 4.60774043885907971676049514492, 5.96492485714835452313233742623, 6.40997546153718178503646187094, 7.58820864190194669731088043629, 8.181778850433677918944686002009, 8.732462468163192598020273793109
