| L(s) = 1 | − 0.660·2-s − 3.19·3-s − 1.56·4-s − 5-s + 2.11·6-s + 2.35·8-s + 7.22·9-s + 0.660·10-s − 0.613·11-s + 5.00·12-s + 2.81·13-s + 3.19·15-s + 1.57·16-s + 7.09·17-s − 4.77·18-s + 6.57·19-s + 1.56·20-s + 0.404·22-s − 4.80·23-s − 7.52·24-s + 25-s − 1.86·26-s − 13.5·27-s + 3.39·29-s − 2.11·30-s − 7.40·31-s − 5.74·32-s + ⋯ |
| L(s) = 1 | − 0.466·2-s − 1.84·3-s − 0.782·4-s − 0.447·5-s + 0.861·6-s + 0.831·8-s + 2.40·9-s + 0.208·10-s − 0.184·11-s + 1.44·12-s + 0.781·13-s + 0.825·15-s + 0.393·16-s + 1.72·17-s − 1.12·18-s + 1.50·19-s + 0.349·20-s + 0.0863·22-s − 1.00·23-s − 1.53·24-s + 0.200·25-s − 0.364·26-s − 2.60·27-s + 0.630·29-s − 0.385·30-s − 1.32·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.660T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 11 | \( 1 + 0.613T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + 0.152T + 53T^{2} \) |
| 59 | \( 1 - 7.42T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 - 4.28T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 - 6.79T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.51243099253062484524628102645, −6.65396795720233290148734117519, −5.88562972325269976287624753932, −5.20538637874411259260800733334, −5.00425035769391328658901874882, −3.86509468668430350903791348694, −3.47002241672170978168176449194, −1.52890308665470170198142280711, −0.943149905974053267478934157163, 0,
0.943149905974053267478934157163, 1.52890308665470170198142280711, 3.47002241672170978168176449194, 3.86509468668430350903791348694, 5.00425035769391328658901874882, 5.20538637874411259260800733334, 5.88562972325269976287624753932, 6.65396795720233290148734117519, 7.51243099253062484524628102645
