| L(s) = 1 | + 2-s − 2·3-s − 4-s + 5-s − 2·6-s + 2·7-s − 3·8-s + 9-s + 10-s + 2·12-s + 13-s + 2·14-s − 2·15-s − 16-s + 5·17-s + 18-s − 20-s − 4·21-s + 2·23-s + 6·24-s − 4·25-s + 26-s + 4·27-s − 2·28-s + 9·29-s − 2·30-s + 2·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s + 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s + 0.769·27-s − 0.377·28-s + 1.67·29-s − 0.365·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.65971922265976, −14.40867344801262, −13.87047669250219, −13.49079680447981, −12.77886431272953, −12.33669489030356, −11.84424675423949, −11.53354455083072, −10.90117530826619, −10.14997767064946, −10.01515903947925, −9.117229409850715, −8.637755575173094, −7.993864806724421, −7.478008567507343, −6.405420897603602, −6.218924384071943, −5.658299454087098, −5.019139449964232, −4.788557233104521, −4.134879869239678, −3.235380587648730, −2.766916759029288, −1.563079450496731, −0.9783039804051472, 0,
0.9783039804051472, 1.563079450496731, 2.766916759029288, 3.235380587648730, 4.134879869239678, 4.788557233104521, 5.019139449964232, 5.658299454087098, 6.218924384071943, 6.405420897603602, 7.478008567507343, 7.993864806724421, 8.637755575173094, 9.117229409850715, 10.01515903947925, 10.14997767064946, 10.90117530826619, 11.53354455083072, 11.84424675423949, 12.33669489030356, 12.77886431272953, 13.49079680447981, 13.87047669250219, 14.40867344801262, 14.65971922265976
