| L(s) = 1 | − 2·2-s + 4·4-s + 4.57·5-s + 7.03·7-s − 8·8-s − 9.15·10-s + 89.4·13-s − 14.0·14-s + 16·16-s − 103.·17-s + 21.0·19-s + 18.3·20-s + 89.9·23-s − 104.·25-s − 178.·26-s + 28.1·28-s − 157.·29-s − 226.·31-s − 32·32-s + 206.·34-s + 32.2·35-s + 174.·37-s − 42.1·38-s − 36.6·40-s − 91.8·41-s + 11.8·43-s − 179.·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.409·5-s + 0.380·7-s − 0.353·8-s − 0.289·10-s + 1.90·13-s − 0.268·14-s + 0.250·16-s − 1.47·17-s + 0.254·19-s + 0.204·20-s + 0.815·23-s − 0.832·25-s − 1.34·26-s + 0.190·28-s − 1.00·29-s − 1.30·31-s − 0.176·32-s + 1.04·34-s + 0.155·35-s + 0.776·37-s − 0.180·38-s − 0.144·40-s − 0.349·41-s + 0.0420·43-s − 0.576·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4.57T + 125T^{2} \) |
| 7 | \( 1 - 7.03T + 343T^{2} \) |
| 13 | \( 1 - 89.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 11.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 122.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 874.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 111.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 813.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 435.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 663.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 204.T + 9.12e5T^{2} \) |
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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
−8.441656146703920119062570451586, −7.71109446976984532553594157072, −6.74643187535923395784435876421, −6.13215958739273241672160284979, −5.31166557386513137260319490941, −4.15738530356690157648931534632, −3.25738633897142277235424784328, −1.99705388344409897099235183576, −1.33643627744202838090858843269, 0,
1.33643627744202838090858843269, 1.99705388344409897099235183576, 3.25738633897142277235424784328, 4.15738530356690157648931534632, 5.31166557386513137260319490941, 6.13215958739273241672160284979, 6.74643187535923395784435876421, 7.71109446976984532553594157072, 8.441656146703920119062570451586
