| L(s) = 1 | + 2.06·3-s + 5-s − 3.39·7-s + 1.26·9-s − 2.27·11-s − 6.00·13-s + 2.06·15-s − 0.0687·17-s + 3.78·19-s − 7.00·21-s + 6.12·23-s + 25-s − 3.58·27-s + 5.66·29-s + 5.02·31-s − 4.70·33-s − 3.39·35-s + 7.20·37-s − 12.4·39-s + 8.51·41-s + 1.14·43-s + 1.26·45-s + 1.28·47-s + 4.51·49-s − 0.141·51-s − 2.53·53-s − 2.27·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s + 0.447·5-s − 1.28·7-s + 0.421·9-s − 0.686·11-s − 1.66·13-s + 0.533·15-s − 0.0166·17-s + 0.867·19-s − 1.52·21-s + 1.27·23-s + 0.200·25-s − 0.689·27-s + 1.05·29-s + 0.901·31-s − 0.818·33-s − 0.573·35-s + 1.18·37-s − 1.98·39-s + 1.32·41-s + 0.175·43-s + 0.188·45-s + 0.187·47-s + 0.644·49-s − 0.0198·51-s − 0.347·53-s − 0.306·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.440341077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.440341077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 0.0687T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 - 5.66T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 37 | \( 1 - 7.20T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 - 8.86T + 61T^{2} \) |
| 67 | \( 1 + 7.20T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 4.14T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 8.37T + 83T^{2} \) |
| 89 | \( 1 - 1.80T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−7.990006770449300289247354039694, −7.48128090896581911731262316729, −6.80754786671155765605529860629, −6.00340773184408500248962887212, −5.15068981167874685072199060875, −4.41418344807528692598397934074, −3.21335750025028316695986636666, −2.81168902919067217198451639626, −2.31672650107475818122819196275, −0.74049359260996754094798392526,
0.74049359260996754094798392526, 2.31672650107475818122819196275, 2.81168902919067217198451639626, 3.21335750025028316695986636666, 4.41418344807528692598397934074, 5.15068981167874685072199060875, 6.00340773184408500248962887212, 6.80754786671155765605529860629, 7.48128090896581911731262316729, 7.990006770449300289247354039694
