| L(s) = 1 | + 2.20·2-s + 3-s + 2.84·4-s + 5-s + 2.20·6-s − 2.98·7-s + 1.85·8-s + 9-s + 2.20·10-s + 6.44·11-s + 2.84·12-s − 4.24·13-s − 6.56·14-s + 15-s − 1.61·16-s + 3.96·17-s + 2.20·18-s + 3.04·19-s + 2.84·20-s − 2.98·21-s + 14.1·22-s + 1.85·24-s + 25-s − 9.34·26-s + 27-s − 8.48·28-s + 7.28·29-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.577·3-s + 1.42·4-s + 0.447·5-s + 0.898·6-s − 1.12·7-s + 0.654·8-s + 0.333·9-s + 0.695·10-s + 1.94·11-s + 0.820·12-s − 1.17·13-s − 1.75·14-s + 0.258·15-s − 0.402·16-s + 0.962·17-s + 0.518·18-s + 0.699·19-s + 0.635·20-s − 0.651·21-s + 3.02·22-s + 0.377·24-s + 0.200·25-s − 1.83·26-s + 0.192·27-s − 1.60·28-s + 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.981229378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.981229378\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 6.44T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 0.662T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 1.19T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 5.35T + 59T^{2} \) |
| 61 | \( 1 - 1.72T + 61T^{2} \) |
| 67 | \( 1 - 7.36T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 9.07T + 97T^{2} \) |
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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
−7.41901091694342790516798276712, −6.95884227834999474525598286973, −6.22717530090721000120625083166, −5.89878107181694438164019153219, −4.79336143818500525710786021962, −4.35025371599585722495144314562, −3.33888399166229897897490732584, −3.13030411548948265165286926945, −2.20352609559880512711449303181, −1.05009439670831447896444044093,
1.05009439670831447896444044093, 2.20352609559880512711449303181, 3.13030411548948265165286926945, 3.33888399166229897897490732584, 4.35025371599585722495144314562, 4.79336143818500525710786021962, 5.89878107181694438164019153219, 6.22717530090721000120625083166, 6.95884227834999474525598286973, 7.41901091694342790516798276712
