| L(s) = 1 | − 4·3-s − 5·5-s − 3·7-s − 11·9-s + 2·11-s − 38·13-s + 20·15-s − 45·17-s + 74·19-s + 12·21-s − 23·23-s + 25·25-s + 152·27-s + 283·29-s + 303·31-s − 8·33-s + 15·35-s + 79·37-s + 152·39-s − 407·41-s + 328·43-s + 55·45-s − 360·47-s − 334·49-s + 180·51-s − 561·53-s − 10·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s − 0.161·7-s − 0.407·9-s + 0.0548·11-s − 0.810·13-s + 0.344·15-s − 0.642·17-s + 0.893·19-s + 0.124·21-s − 0.208·23-s + 1/5·25-s + 1.08·27-s + 1.81·29-s + 1.75·31-s − 0.0422·33-s + 0.0724·35-s + 0.351·37-s + 0.624·39-s − 1.55·41-s + 1.16·43-s + 0.182·45-s − 1.11·47-s − 0.973·49-s + 0.494·51-s − 1.45·53-s − 0.0245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 45 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 - 283 T + p^{3} T^{2} \) |
| 31 | \( 1 - 303 T + p^{3} T^{2} \) |
| 37 | \( 1 - 79 T + p^{3} T^{2} \) |
| 41 | \( 1 + 407 T + p^{3} T^{2} \) |
| 43 | \( 1 - 328 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 561 T + p^{3} T^{2} \) |
| 59 | \( 1 + 101 T + p^{3} T^{2} \) |
| 61 | \( 1 + 268 T + p^{3} T^{2} \) |
| 67 | \( 1 - 69 T + p^{3} T^{2} \) |
| 71 | \( 1 - 641 T + p^{3} T^{2} \) |
| 73 | \( 1 - 994 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + 503 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1608 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1082 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.350353852069957397219518833636, −7.80474427185014830172932358282, −6.60100693834804215948856230943, −6.34240357455061918242452016224, −4.99576676805502991450970103548, −4.73820242050662571764194676859, −3.36157132471461982854474163153, −2.52168475873601553448366536592, −0.979006572399322826682304997498, 0,
0.979006572399322826682304997498, 2.52168475873601553448366536592, 3.36157132471461982854474163153, 4.73820242050662571764194676859, 4.99576676805502991450970103548, 6.34240357455061918242452016224, 6.60100693834804215948856230943, 7.80474427185014830172932358282, 8.350353852069957397219518833636
