| L(s) = 1 | + 3-s + 2.44·7-s + 9-s − 0.576·11-s + 6.44·13-s + 1.89·17-s + 8.27·19-s + 2.44·21-s + 4.20·23-s + 27-s + 6.53·29-s + 4.86·31-s − 0.576·33-s − 0.218·37-s + 6.44·39-s + 6.45·41-s − 3.42·43-s − 9.61·47-s − 1.02·49-s + 1.89·51-s − 13.9·53-s + 8.27·57-s − 12.0·59-s − 4.81·61-s + 2.44·63-s + 3.87·67-s + 4.20·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.923·7-s + 0.333·9-s − 0.173·11-s + 1.78·13-s + 0.460·17-s + 1.89·19-s + 0.533·21-s + 0.877·23-s + 0.192·27-s + 1.21·29-s + 0.874·31-s − 0.100·33-s − 0.0358·37-s + 1.03·39-s + 1.00·41-s − 0.522·43-s − 1.40·47-s − 0.146·49-s + 0.265·51-s − 1.92·53-s + 1.09·57-s − 1.56·59-s − 0.615·61-s + 0.307·63-s + 0.472·67-s + 0.506·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.771819967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.771819967\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 0.576T + 11T^{2} \) |
| 13 | \( 1 - 6.44T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 8.27T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 0.218T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 9.61T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
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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
−8.101825110101404443639240268899, −7.36866690582519798052497245349, −6.50821621052406932053305297281, −5.80701596147050436162410004509, −4.94837054682698674512124896718, −4.40837845533020150814287706369, −3.22975718435071746987491430215, −3.03090422401940533277026086095, −1.51731785227298098594008721923, −1.13353965148705270727676309968,
1.13353965148705270727676309968, 1.51731785227298098594008721923, 3.03090422401940533277026086095, 3.22975718435071746987491430215, 4.40837845533020150814287706369, 4.94837054682698674512124896718, 5.80701596147050436162410004509, 6.50821621052406932053305297281, 7.36866690582519798052497245349, 8.101825110101404443639240268899
