L(s) = 1 | − 3-s + 3·5-s − 7-s + 9-s + 3·11-s + 2·13-s − 3·15-s − 2·19-s + 21-s + 4·25-s − 27-s + 3·29-s − 5·31-s − 3·33-s − 3·35-s + 2·37-s − 2·39-s + 10·43-s + 3·45-s − 12·47-s + 49-s + 3·53-s + 9·55-s + 2·57-s − 3·59-s − 4·61-s − 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.554·13-s − 0.774·15-s − 0.458·19-s + 0.218·21-s + 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.522·33-s − 0.507·35-s + 0.328·37-s − 0.320·39-s + 1.52·43-s + 0.447·45-s − 1.75·47-s + 1/7·49-s + 0.412·53-s + 1.21·55-s + 0.264·57-s − 0.390·59-s − 0.512·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.91318087650935, −13.55652277426928, −13.04120890772559, −12.58450666251732, −12.21914643007694, −11.42484546869800, −11.13527924663869, −10.55802595051953, −10.04018250162475, −9.635656232485590, −9.149498354592915, −8.752368650968699, −8.063699872906607, −7.327319236482690, −6.753190312100471, −6.285560364654245, −6.012254166892535, −5.457908667295411, −4.861241239623797, −4.174018824974588, −3.665477888022159, −2.870742222082384, −2.211116104279973, −1.524351820215818, −1.056252472671250, 0,
1.056252472671250, 1.524351820215818, 2.211116104279973, 2.870742222082384, 3.665477888022159, 4.174018824974588, 4.861241239623797, 5.457908667295411, 6.012254166892535, 6.285560364654245, 6.753190312100471, 7.327319236482690, 8.063699872906607, 8.752368650968699, 9.149498354592915, 9.635656232485590, 10.04018250162475, 10.55802595051953, 11.13527924663869, 11.42484546869800, 12.21914643007694, 12.58450666251732, 13.04120890772559, 13.55652277426928, 13.91318087650935