| L(s) = 1 | + 0.730·2-s − 6.29·3-s − 7.46·4-s − 4.59·6-s + 0.329·7-s − 11.2·8-s + 12.5·9-s − 35.1·11-s + 46.9·12-s + 27.3·13-s + 0.240·14-s + 51.4·16-s + 3.19·17-s + 9.17·18-s − 107.·19-s − 2.07·21-s − 25.6·22-s − 99.8·23-s + 71.0·24-s + 19.9·26-s + 90.7·27-s − 2.46·28-s + 25.4·29-s + 164.·31-s + 127.·32-s + 220.·33-s + 2.33·34-s + ⋯ |
| L(s) = 1 | + 0.258·2-s − 1.21·3-s − 0.933·4-s − 0.312·6-s + 0.0178·7-s − 0.499·8-s + 0.465·9-s − 0.962·11-s + 1.13·12-s + 0.583·13-s + 0.00459·14-s + 0.804·16-s + 0.0455·17-s + 0.120·18-s − 1.29·19-s − 0.0215·21-s − 0.248·22-s − 0.905·23-s + 0.604·24-s + 0.150·26-s + 0.646·27-s − 0.0166·28-s + 0.163·29-s + 0.951·31-s + 0.706·32-s + 1.16·33-s + 0.0117·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 71 | \( 1 - 71T \) |
| good | 2 | \( 1 - 0.730T + 8T^{2} \) |
| 3 | \( 1 + 6.29T + 27T^{2} \) |
| 7 | \( 1 - 0.329T + 343T^{2} \) |
| 11 | \( 1 + 35.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 3.19T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 25.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 202.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 378.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 381.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 792.T + 3.00e5T^{2} \) |
| 73 | \( 1 + 793.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 825.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 777.T + 9.12e5T^{2} \) |
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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
−8.362192506670255036472497226163, −7.979017956223657761536610346175, −6.53347836935100210935201358750, −6.04326510384266873055461398492, −5.23436526203413768229090069939, −4.59860391721344144581165440697, −3.74555808449132126014629360350, −2.45358247933192375272999722949, −0.860412856381953324714227254857, 0,
0.860412856381953324714227254857, 2.45358247933192375272999722949, 3.74555808449132126014629360350, 4.59860391721344144581165440697, 5.23436526203413768229090069939, 6.04326510384266873055461398492, 6.53347836935100210935201358750, 7.979017956223657761536610346175, 8.362192506670255036472497226163
