| L(s) = 1 | − 2·7-s + 11-s − 4·13-s − 2·17-s + 23-s − 7·31-s − 3·37-s + 8·41-s − 6·43-s − 8·47-s − 3·49-s − 6·53-s + 5·59-s + 12·61-s − 7·67-s − 3·71-s − 4·73-s − 2·77-s + 10·79-s + 6·83-s − 15·89-s + 8·91-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 0.208·23-s − 1.25·31-s − 0.493·37-s + 1.24·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.650·59-s + 1.53·61-s − 0.855·67-s − 0.356·71-s − 0.468·73-s − 0.227·77-s + 1.12·79-s + 0.658·83-s − 1.58·89-s + 0.838·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8192710830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8192710830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.75926281733095, −14.41111458083851, −13.73843017448697, −13.10378771850520, −12.76894349467224, −12.30790372749137, −11.62293216446684, −11.21026181058708, −10.54756065599058, −9.966787628479209, −9.485833426162263, −9.123988494128151, −8.399321207327632, −7.799299595540175, −7.117986170101908, −6.757004810682732, −6.165305201768287, −5.413257902950058, −4.924042694674779, −4.198468048056038, −3.551345965049679, −2.918238330464930, −2.229013145737167, −1.473965456016027, −0.3226584835751798,
0.3226584835751798, 1.473965456016027, 2.229013145737167, 2.918238330464930, 3.551345965049679, 4.198468048056038, 4.924042694674779, 5.413257902950058, 6.165305201768287, 6.757004810682732, 7.117986170101908, 7.799299595540175, 8.399321207327632, 9.123988494128151, 9.485833426162263, 9.966787628479209, 10.54756065599058, 11.21026181058708, 11.62293216446684, 12.30790372749137, 12.76894349467224, 13.10378771850520, 13.73843017448697, 14.41111458083851, 14.75926281733095
