| L(s) = 1 | + 0.471·3-s + 0.567·7-s − 2.77·9-s + 4.37·11-s − 0.165·13-s − 7.94·17-s + 19-s + 0.267·21-s + 3.87·23-s − 2.72·27-s + 3.53·29-s + 3.20·31-s + 2.06·33-s + 10.1·37-s − 0.0779·39-s − 5.97·41-s + 12.0·43-s − 5.46·47-s − 6.67·49-s − 3.74·51-s − 2.00·53-s + 0.471·57-s + 8.32·59-s + 11.8·61-s − 1.57·63-s + 8.79·67-s + 1.82·69-s + ⋯ |
| L(s) = 1 | + 0.271·3-s + 0.214·7-s − 0.926·9-s + 1.32·11-s − 0.0458·13-s − 1.92·17-s + 0.229·19-s + 0.0583·21-s + 0.807·23-s − 0.523·27-s + 0.655·29-s + 0.575·31-s + 0.358·33-s + 1.67·37-s − 0.0124·39-s − 0.932·41-s + 1.84·43-s − 0.796·47-s − 0.954·49-s − 0.524·51-s − 0.275·53-s + 0.0623·57-s + 1.08·59-s + 1.51·61-s − 0.198·63-s + 1.07·67-s + 0.219·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033154134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033154134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.471T + 3T^{2} \) |
| 7 | \( 1 - 0.567T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 0.165T + 13T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 - 3.20T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.97T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 + 2.00T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 0.720T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.500193341886418798288013217090, −7.984820849934032354252099659200, −6.70582164544817996685952791504, −6.58743311623019396398719591619, −5.50809319115833067201241661972, −4.60314274779459705053605964331, −3.93283401751500165448886812878, −2.89417324206712532907744656805, −2.11319549206465084375759513915, −0.818379065893886754149673685414,
0.818379065893886754149673685414, 2.11319549206465084375759513915, 2.89417324206712532907744656805, 3.93283401751500165448886812878, 4.60314274779459705053605964331, 5.50809319115833067201241661972, 6.58743311623019396398719591619, 6.70582164544817996685952791504, 7.984820849934032354252099659200, 8.500193341886418798288013217090
