| L(s) = 1 | − 19·5-s − 13·7-s − 65·11-s + 56·13-s + 108·17-s + 58·19-s − 66·23-s + 236·25-s − 118·29-s + 145·31-s + 247·35-s − 190·37-s + 430·41-s + 530·43-s − 74·47-s − 174·49-s + 295·53-s + 1.23e3·55-s − 628·59-s − 360·61-s − 1.06e3·65-s + 146·67-s + 388·71-s + 753·73-s + 845·77-s − 1.13e3·79-s − 153·83-s + ⋯ |
| L(s) = 1 | − 1.69·5-s − 0.701·7-s − 1.78·11-s + 1.19·13-s + 1.54·17-s + 0.700·19-s − 0.598·23-s + 1.88·25-s − 0.755·29-s + 0.840·31-s + 1.19·35-s − 0.844·37-s + 1.63·41-s + 1.87·43-s − 0.229·47-s − 0.507·49-s + 0.764·53-s + 3.02·55-s − 1.38·59-s − 0.755·61-s − 2.03·65-s + 0.266·67-s + 0.648·71-s + 1.20·73-s + 1.25·77-s − 1.61·79-s − 0.202·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 65 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 - 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 58 T + p^{3} T^{2} \) |
| 23 | \( 1 + 66 T + p^{3} T^{2} \) |
| 29 | \( 1 + 118 T + p^{3} T^{2} \) |
| 31 | \( 1 - 145 T + p^{3} T^{2} \) |
| 37 | \( 1 + 190 T + p^{3} T^{2} \) |
| 41 | \( 1 - 430 T + p^{3} T^{2} \) |
| 43 | \( 1 - 530 T + p^{3} T^{2} \) |
| 47 | \( 1 + 74 T + p^{3} T^{2} \) |
| 53 | \( 1 - 295 T + p^{3} T^{2} \) |
| 59 | \( 1 + 628 T + p^{3} T^{2} \) |
| 61 | \( 1 + 360 T + p^{3} T^{2} \) |
| 67 | \( 1 - 146 T + p^{3} T^{2} \) |
| 71 | \( 1 - 388 T + p^{3} T^{2} \) |
| 73 | \( 1 - 753 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 153 T + p^{3} T^{2} \) |
| 89 | \( 1 + 850 T + p^{3} T^{2} \) |
| 97 | \( 1 - 391 T + p^{3} T^{2} \) |
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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
−8.267228688458392268061716094056, −7.78521159665853743899121770635, −7.32187358401920481097732464653, −6.05689708013269702406256261464, −5.33382750618740795131717207593, −4.20487425707028616535278384177, −3.44238459676677560475912160530, −2.80275419837552357000615512934, −0.963905348462018416393324388523, 0,
0.963905348462018416393324388523, 2.80275419837552357000615512934, 3.44238459676677560475912160530, 4.20487425707028616535278384177, 5.33382750618740795131717207593, 6.05689708013269702406256261464, 7.32187358401920481097732464653, 7.78521159665853743899121770635, 8.267228688458392268061716094056
