| L(s) = 1 | − 0.487·2-s + 3-s − 1.76·4-s + 1.53·5-s − 0.487·6-s + 1.83·8-s + 9-s − 0.747·10-s − 3.51·11-s − 1.76·12-s + 2.32·13-s + 1.53·15-s + 2.62·16-s − 4.08·17-s − 0.487·18-s + 1.32·19-s − 2.69·20-s + 1.71·22-s + 1.38·23-s + 1.83·24-s − 2.65·25-s − 1.13·26-s + 27-s − 5.15·29-s − 0.747·30-s + 0.813·31-s − 4.95·32-s + ⋯ |
| L(s) = 1 | − 0.345·2-s + 0.577·3-s − 0.880·4-s + 0.685·5-s − 0.199·6-s + 0.648·8-s + 0.333·9-s − 0.236·10-s − 1.05·11-s − 0.508·12-s + 0.645·13-s + 0.395·15-s + 0.657·16-s − 0.991·17-s − 0.115·18-s + 0.302·19-s − 0.603·20-s + 0.365·22-s + 0.289·23-s + 0.374·24-s − 0.530·25-s − 0.222·26-s + 0.192·27-s − 0.957·29-s − 0.136·30-s + 0.146·31-s − 0.875·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.487T + 2T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 - 0.813T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 43 | \( 1 + 2.79T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 8.05T + 53T^{2} \) |
| 59 | \( 1 - 0.693T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.052352830868739646601299251094, −7.19664239580488711055781648788, −6.27342658580401029988554685352, −5.51475045642441513929165206408, −4.81784686711234850803665036821, −4.06498196738934487574372100946, −3.18117175912250409853970109967, −2.25332696127074385487073776648, −1.35707700099608622104233391786, 0,
1.35707700099608622104233391786, 2.25332696127074385487073776648, 3.18117175912250409853970109967, 4.06498196738934487574372100946, 4.81784686711234850803665036821, 5.51475045642441513929165206408, 6.27342658580401029988554685352, 7.19664239580488711055781648788, 8.052352830868739646601299251094
