| L(s) = 1 | + 3-s − 7-s + 9-s − 13-s + 6·17-s − 19-s − 21-s + 8·23-s + 27-s + 7·29-s − 10·31-s + 3·37-s − 39-s − 4·43-s − 7·47-s + 49-s + 6·51-s − 4·53-s − 57-s + 7·59-s + 14·61-s − 63-s + 13·67-s + 8·69-s − 16·71-s + 13·73-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 1.45·17-s − 0.229·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s + 1.29·29-s − 1.79·31-s + 0.493·37-s − 0.160·39-s − 0.609·43-s − 1.02·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.132·57-s + 0.911·59-s + 1.79·61-s − 0.125·63-s + 1.58·67-s + 0.963·69-s − 1.89·71-s + 1.52·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.740728009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.740728009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.73996420010780, −12.65804918280386, −11.91778710683687, −11.50037715710414, −10.94150245342503, −10.49104210248752, −9.897033883316854, −9.667403925981310, −9.146638912793371, −8.616988270689857, −8.181023866737997, −7.733498001864964, −7.111460607766578, −6.827088605443454, −6.298220146782865, −5.490627438358383, −5.226082378910356, −4.678800888648237, −3.927799127633008, −3.459562371002913, −3.046641461447625, −2.503936817635048, −1.813311053383331, −1.125483176395599, −0.5506924370801580,
0.5506924370801580, 1.125483176395599, 1.813311053383331, 2.503936817635048, 3.046641461447625, 3.459562371002913, 3.927799127633008, 4.678800888648237, 5.226082378910356, 5.490627438358383, 6.298220146782865, 6.827088605443454, 7.111460607766578, 7.733498001864964, 8.181023866737997, 8.616988270689857, 9.146638912793371, 9.667403925981310, 9.897033883316854, 10.49104210248752, 10.94150245342503, 11.50037715710414, 11.91778710683687, 12.65804918280386, 12.73996420010780
