| L(s) = 1 | + 2-s + 2.44·3-s + 4-s + 2.44·6-s + 8-s + 2.99·9-s + 4.89·11-s + 2.44·12-s − 0.449·13-s + 16-s + 2·17-s + 2.99·18-s − 6.44·19-s + 4.89·22-s + 6.89·23-s + 2.44·24-s − 0.449·26-s − 2.89·29-s + 0.898·31-s + 32-s + 11.9·33-s + 2·34-s + 2.99·36-s − 2·37-s − 6.44·38-s − 1.10·39-s + 10.8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.999·6-s + 0.353·8-s + 0.999·9-s + 1.47·11-s + 0.707·12-s − 0.124·13-s + 0.250·16-s + 0.485·17-s + 0.707·18-s − 1.47·19-s + 1.04·22-s + 1.43·23-s + 0.499·24-s − 0.0881·26-s − 0.538·29-s + 0.161·31-s + 0.176·32-s + 2.08·33-s + 0.342·34-s + 0.499·36-s − 0.328·37-s − 1.04·38-s − 0.176·39-s + 1.70·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.029900059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.029900059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 - 6.89T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 0.898T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 0.898T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−8.927698328760850481354312932010, −8.260994610730353670738771461189, −7.35577307783285726690172344549, −6.71352982921670967095350357854, −5.87899271979687036458120926247, −4.69220445963268926386245835756, −3.94736337617112802129572639890, −3.28636743830996252219875624305, −2.38452022263727889040219876873, −1.41479414346004745625551718933,
1.41479414346004745625551718933, 2.38452022263727889040219876873, 3.28636743830996252219875624305, 3.94736337617112802129572639890, 4.69220445963268926386245835756, 5.87899271979687036458120926247, 6.71352982921670967095350357854, 7.35577307783285726690172344549, 8.260994610730353670738771461189, 8.927698328760850481354312932010
