| L(s) = 1 | − 2.67·2-s − 2.48·3-s + 5.15·4-s − 5-s + 6.63·6-s − 7-s − 8.44·8-s + 3.15·9-s + 2.67·10-s − 11-s − 12.7·12-s + 5.83·13-s + 2.67·14-s + 2.48·15-s + 12.2·16-s + 5.44·17-s − 8.44·18-s − 1.35·19-s − 5.15·20-s + 2.48·21-s + 2.67·22-s − 3.19·23-s + 20.9·24-s + 25-s − 15.5·26-s − 0.387·27-s − 5.15·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.43·3-s + 2.57·4-s − 0.447·5-s + 2.70·6-s − 0.377·7-s − 2.98·8-s + 1.05·9-s + 0.845·10-s − 0.301·11-s − 3.69·12-s + 1.61·13-s + 0.714·14-s + 0.640·15-s + 3.06·16-s + 1.32·17-s − 1.99·18-s − 0.309·19-s − 1.15·20-s + 0.541·21-s + 0.570·22-s − 0.665·23-s + 4.27·24-s + 0.200·25-s − 3.05·26-s − 0.0746·27-s − 0.974·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 + 2.48T + 3T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 0.231T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.22T + 89T^{2} \) |
| 97 | \( 1 + 0.836T + 97T^{2} \) |
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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
−10.61731617634638183911717946128, −10.26225398156256721273062552318, −9.055476552138179875249008093541, −8.170616216374262506385874503087, −7.22848268834463671141869238016, −6.29788985528198938684929791788, −5.57552992287020979104674789916, −3.48355456162881564220919586003, −1.41072267064557541249778227384, 0,
1.41072267064557541249778227384, 3.48355456162881564220919586003, 5.57552992287020979104674789916, 6.29788985528198938684929791788, 7.22848268834463671141869238016, 8.170616216374262506385874503087, 9.055476552138179875249008093541, 10.26225398156256721273062552318, 10.61731617634638183911717946128
