| L(s) = 1 | + 11-s + 5·13-s − 6·17-s + 7·19-s − 4·23-s − 5·25-s − 2·29-s + 7·31-s − 7·37-s − 4·41-s − 9·43-s + 6·47-s − 2·53-s − 12·59-s + 2·61-s + 7·67-s + 8·71-s − 5·73-s + 11·79-s + 4·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.38·13-s − 1.45·17-s + 1.60·19-s − 0.834·23-s − 25-s − 0.371·29-s + 1.25·31-s − 1.15·37-s − 0.624·41-s − 1.37·43-s + 0.875·47-s − 0.274·53-s − 1.56·59-s + 0.256·61-s + 0.855·67-s + 0.949·71-s − 0.585·73-s + 1.23·79-s + 0.439·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.429768639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.429768639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.64613260109444, −12.02991806291662, −11.76977784266240, −11.31794769306900, −10.93177916874605, −10.35798776293836, −9.894859569689054, −9.460604945697237, −8.966818501837646, −8.450267944916509, −8.178496945760716, −7.548061126598210, −7.031374609468413, −6.516054108141490, −6.122685269935381, −5.677344479677911, −5.034541343202070, −4.546532666386478, −3.991379245429768, −3.366855850671039, −3.216025208207216, −2.104283094606155, −1.877947058385635, −1.109560413770215, −0.4387387545307603,
0.4387387545307603, 1.109560413770215, 1.877947058385635, 2.104283094606155, 3.216025208207216, 3.366855850671039, 3.991379245429768, 4.546532666386478, 5.034541343202070, 5.677344479677911, 6.122685269935381, 6.516054108141490, 7.031374609468413, 7.548061126598210, 8.178496945760716, 8.450267944916509, 8.966818501837646, 9.460604945697237, 9.894859569689054, 10.35798776293836, 10.93177916874605, 11.31794769306900, 11.76977784266240, 12.02991806291662, 12.64613260109444
