| L(s) = 1 | + 2-s − 3-s + 4-s − 0.435·5-s − 6-s + 1.91·7-s + 8-s + 9-s − 0.435·10-s + 2.85·11-s − 12-s + 13-s + 1.91·14-s + 0.435·15-s + 16-s − 3.59·17-s + 18-s − 3.69·19-s − 0.435·20-s − 1.91·21-s + 2.85·22-s − 1.86·23-s − 24-s − 4.81·25-s + 26-s − 27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.194·5-s − 0.408·6-s + 0.725·7-s + 0.353·8-s + 0.333·9-s − 0.137·10-s + 0.860·11-s − 0.288·12-s + 0.277·13-s + 0.513·14-s + 0.112·15-s + 0.250·16-s − 0.870·17-s + 0.235·18-s − 0.847·19-s − 0.0973·20-s − 0.418·21-s + 0.608·22-s − 0.389·23-s − 0.204·24-s − 0.962·25-s + 0.196·26-s − 0.192·27-s + 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.435T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 0.961T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 9.27T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 0.983T + 83T^{2} \) |
| 89 | \( 1 - 2.06T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−7.37212585745091756067945254745, −6.50370573268596777021166910531, −6.21391251541404017162503358494, −5.30280589492662240510948131217, −4.62900232908185095175059091819, −4.07288590048521428900307756796, −3.36667974088197718461359505387, −2.06039872397322528046770878509, −1.52145066216255556474930602595, 0,
1.52145066216255556474930602595, 2.06039872397322528046770878509, 3.36667974088197718461359505387, 4.07288590048521428900307756796, 4.62900232908185095175059091819, 5.30280589492662240510948131217, 6.21391251541404017162503358494, 6.50370573268596777021166910531, 7.37212585745091756067945254745
