| L(s) = 1 | + 3·3-s − 2·4-s + 4·5-s + 7-s + 6·9-s + 11-s − 6·12-s + 4·13-s + 12·15-s + 4·16-s − 2·17-s − 8·20-s + 3·21-s − 8·23-s + 11·25-s + 9·27-s − 2·28-s + 2·31-s + 3·33-s + 4·35-s − 12·36-s + 37-s + 12·39-s + 3·41-s + 4·43-s − 2·44-s + 24·45-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s + 1.78·5-s + 0.377·7-s + 2·9-s + 0.301·11-s − 1.73·12-s + 1.10·13-s + 3.09·15-s + 16-s − 0.485·17-s − 1.78·20-s + 0.654·21-s − 1.66·23-s + 11/5·25-s + 1.73·27-s − 0.377·28-s + 0.359·31-s + 0.522·33-s + 0.676·35-s − 2·36-s + 0.164·37-s + 1.92·39-s + 0.468·41-s + 0.609·43-s − 0.301·44-s + 3.57·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.046059641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.046059641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 163 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.351771356097876969660425035011, −7.71680613030940579451629449923, −6.55595938132286923918154384002, −6.00316218377003405435128311636, −5.12095287776129405390715460793, −4.25124999233486318475601475852, −3.64370137113890380487368126513, −2.68203080826265037078350715700, −1.90165334389356944710043750517, −1.25469741854338938830336054671,
1.25469741854338938830336054671, 1.90165334389356944710043750517, 2.68203080826265037078350715700, 3.64370137113890380487368126513, 4.25124999233486318475601475852, 5.12095287776129405390715460793, 6.00316218377003405435128311636, 6.55595938132286923918154384002, 7.71680613030940579451629449923, 8.351771356097876969660425035011
