| L(s) = 1 | + 1.59·3-s − 4.51·7-s − 0.460·9-s + 0.593·11-s + 4.05·13-s + 5.32·17-s + 19-s − 7.19·21-s + 7.46·23-s − 5.51·27-s − 2.32·29-s + 6.97·31-s + 0.945·33-s − 3.40·37-s + 6.46·39-s − 7.38·41-s + 6.18·43-s + 12.7·47-s + 13.3·49-s + 8.48·51-s − 8.84·53-s + 1.59·57-s + 4.10·59-s + 9.89·61-s + 2.07·63-s + 12.0·67-s + 11.8·69-s + ⋯ |
| L(s) = 1 | + 0.920·3-s − 1.70·7-s − 0.153·9-s + 0.178·11-s + 1.12·13-s + 1.29·17-s + 0.229·19-s − 1.56·21-s + 1.55·23-s − 1.06·27-s − 0.432·29-s + 1.25·31-s + 0.164·33-s − 0.560·37-s + 1.03·39-s − 1.15·41-s + 0.943·43-s + 1.85·47-s + 1.91·49-s + 1.18·51-s − 1.21·53-s + 0.211·57-s + 0.534·59-s + 1.26·61-s + 0.261·63-s + 1.47·67-s + 1.43·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.087843089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.087843089\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.59T + 3T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 - 0.593T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 - 4.10T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 + 3.96T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
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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
−9.133830282234909634770578790623, −8.630452090477568030155337166545, −7.71204780021486835538272686341, −6.85115601380819558202290901603, −6.12113002595069626494772234844, −5.31266877318578960520081929516, −3.76540006699387138889757868476, −3.34981108553319591268307388853, −2.58735839504172990475993803485, −0.951754256663920820649953696049,
0.951754256663920820649953696049, 2.58735839504172990475993803485, 3.34981108553319591268307388853, 3.76540006699387138889757868476, 5.31266877318578960520081929516, 6.12113002595069626494772234844, 6.85115601380819558202290901603, 7.71204780021486835538272686341, 8.630452090477568030155337166545, 9.133830282234909634770578790623
