L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s + 13-s + 14-s + 16-s − 6·17-s + 18-s − 4·19-s + 21-s + 24-s − 5·25-s + 26-s + 27-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s + 2·37-s − 4·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 41 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.64544263932852, −15.19479800168627, −14.68316658117965, −14.11059201919170, −13.66564396805786, −13.13289330209126, −12.73151623446536, −12.03311742173739, −11.43159115808876, −10.95624344407962, −10.40191043305114, −9.746197554744651, −8.990636678403984, −8.538879946870052, −7.956073916709192, −7.328874966948356, −6.572361096151101, −6.239830889975726, −5.382907272327789, −4.603235818263475, −4.245219636934777, −3.540357619780032, −2.715647435777059, −2.097585733680272, −1.430836187473806, 0,
1.430836187473806, 2.097585733680272, 2.715647435777059, 3.540357619780032, 4.245219636934777, 4.603235818263475, 5.382907272327789, 6.239830889975726, 6.572361096151101, 7.328874966948356, 7.956073916709192, 8.538879946870052, 8.990636678403984, 9.746197554744651, 10.40191043305114, 10.95624344407962, 11.43159115808876, 12.03311742173739, 12.73151623446536, 13.13289330209126, 13.66564396805786, 14.11059201919170, 14.68316658117965, 15.19479800168627, 15.64544263932852