| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 12-s + 4·13-s − 4·14-s + 16-s − 6·17-s − 18-s + 19-s − 4·21-s + 6·23-s + 24-s − 4·26-s − 27-s + 4·28-s + 6·29-s + 2·31-s − 32-s + 6·34-s + 36-s + 4·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.872·21-s + 1.25·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.657·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408217793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408217793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.704050932838352025137892184521, −8.159105355683612401574240174584, −7.35504934788273792149934367553, −6.56150532808784242280635540516, −5.85314026027137857149722844402, −4.82634328600464260130703153237, −4.31166193556273879958873044536, −2.88557183348354831390313776689, −1.73257426587651232216444511918, −0.903533601883457432378347653438,
0.903533601883457432378347653438, 1.73257426587651232216444511918, 2.88557183348354831390313776689, 4.31166193556273879958873044536, 4.82634328600464260130703153237, 5.85314026027137857149722844402, 6.56150532808784242280635540516, 7.35504934788273792149934367553, 8.159105355683612401574240174584, 8.704050932838352025137892184521
