L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s + 15-s + 16-s − 2·17-s + 18-s + 19-s + 20-s + 4·22-s + 4·23-s + 24-s + 25-s − 2·26-s + 27-s + 6·29-s + 30-s − 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.119386905\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.119386905\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.03547814963810, −14.65806648301672, −14.15187963538919, −13.61933167619488, −13.34833875615457, −12.53716964018144, −12.12527671107631, −11.67439655103248, −10.92028009989562, −10.39949480740590, −9.762334952395729, −9.249176122900264, −8.653738967147683, −8.203119060569047, −7.134780285397171, −6.915982857711752, −6.441282666602501, −5.434108622759854, −5.128662381210720, −4.265677489386675, −3.757442976020361, −3.074301387431393, −2.346267683996960, −1.719468244419001, −0.8470253780428891,
0.8470253780428891, 1.719468244419001, 2.346267683996960, 3.074301387431393, 3.757442976020361, 4.265677489386675, 5.128662381210720, 5.434108622759854, 6.441282666602501, 6.915982857711752, 7.134780285397171, 8.203119060569047, 8.653738967147683, 9.249176122900264, 9.762334952395729, 10.39949480740590, 10.92028009989562, 11.67439655103248, 12.12527671107631, 12.53716964018144, 13.34833875615457, 13.61933167619488, 14.15187963538919, 14.65806648301672, 15.03547814963810