| L(s) = 1 | + 0.816·2-s + 1.53·3-s − 1.33·4-s − 5-s + 1.25·6-s + 5.03·7-s − 2.72·8-s − 0.633·9-s − 0.816·10-s − 3.03·11-s − 2.05·12-s − 4.57·13-s + 4.11·14-s − 1.53·15-s + 0.443·16-s − 1.07·17-s − 0.517·18-s + 19-s + 1.33·20-s + 7.74·21-s − 2.47·22-s + 4.11·23-s − 4.18·24-s + 25-s − 3.73·26-s − 5.58·27-s − 6.71·28-s + ⋯ |
| L(s) = 1 | + 0.577·2-s + 0.888·3-s − 0.666·4-s − 0.447·5-s + 0.512·6-s + 1.90·7-s − 0.962·8-s − 0.211·9-s − 0.258·10-s − 0.914·11-s − 0.592·12-s − 1.26·13-s + 1.09·14-s − 0.397·15-s + 0.110·16-s − 0.261·17-s − 0.121·18-s + 0.229·19-s + 0.298·20-s + 1.68·21-s − 0.528·22-s + 0.857·23-s − 0.854·24-s + 0.200·25-s − 0.732·26-s − 1.07·27-s − 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375636158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375636158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.816T + 2T^{2} \) |
| 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 0.0947T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 - 5.28T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 + 2.16T + 97T^{2} \) |
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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
−14.26130826641689113741472137633, −13.18340986649596871940767958234, −12.01852929511958864368013333007, −10.98458192117391506294016079182, −9.389715201950502876245592725518, −8.276465298966013527834539267026, −7.68669710050338327374469583993, −5.27426738456968623481319456299, −4.47077755270593959731533655099, −2.70606734603912392472615532973,
2.70606734603912392472615532973, 4.47077755270593959731533655099, 5.27426738456968623481319456299, 7.68669710050338327374469583993, 8.276465298966013527834539267026, 9.389715201950502876245592725518, 10.98458192117391506294016079182, 12.01852929511958864368013333007, 13.18340986649596871940767958234, 14.26130826641689113741472137633
