| L(s) = 1 | + 0.610·2-s − 0.500·3-s − 1.62·4-s + 5-s − 0.305·6-s − 7-s − 2.21·8-s − 2.74·9-s + 0.610·10-s − 0.366·11-s + 0.814·12-s − 5.87·13-s − 0.610·14-s − 0.500·15-s + 1.90·16-s + 5.46·17-s − 1.67·18-s + 5.46·19-s − 1.62·20-s + 0.500·21-s − 0.224·22-s − 2.65·23-s + 1.10·24-s + 25-s − 3.59·26-s + 2.87·27-s + 1.62·28-s + ⋯ |
| L(s) = 1 | + 0.431·2-s − 0.289·3-s − 0.813·4-s + 0.447·5-s − 0.124·6-s − 0.377·7-s − 0.783·8-s − 0.916·9-s + 0.193·10-s − 0.110·11-s + 0.235·12-s − 1.63·13-s − 0.163·14-s − 0.129·15-s + 0.475·16-s + 1.32·17-s − 0.395·18-s + 1.25·19-s − 0.363·20-s + 0.109·21-s − 0.0477·22-s − 0.552·23-s + 0.226·24-s + 0.200·25-s − 0.704·26-s + 0.554·27-s + 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.610T + 2T^{2} \) |
| 3 | \( 1 + 0.500T + 3T^{2} \) |
| 11 | \( 1 + 0.366T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 + 0.818T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 9.13T + 47T^{2} \) |
| 53 | \( 1 - 5.70T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−7.57820862053972571289071852038, −6.64547916372258697530376861697, −5.76167923533061795408821907496, −5.44708096135527931015413052936, −4.86670804427775284914475203161, −3.96417034430648967517324746616, −3.01507261594413959667818790951, −2.60498179734315313775950833976, −1.05185284869949612151728450335, 0,
1.05185284869949612151728450335, 2.60498179734315313775950833976, 3.01507261594413959667818790951, 3.96417034430648967517324746616, 4.86670804427775284914475203161, 5.44708096135527931015413052936, 5.76167923533061795408821907496, 6.64547916372258697530376861697, 7.57820862053972571289071852038
