| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 6·11-s − 13-s + 15-s − 17-s + 2·19-s + 2·21-s + 6·23-s + 25-s − 27-s − 9·29-s − 9·31-s − 6·33-s + 2·35-s − 12·37-s + 39-s + 6·41-s − 6·43-s − 45-s + 12·47-s − 3·49-s + 51-s − 3·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 1.61·31-s − 1.04·33-s + 0.338·35-s − 1.97·37-s + 0.160·39-s + 0.937·41-s − 0.914·43-s − 0.149·45-s + 1.75·47-s − 3/7·49-s + 0.140·51-s − 0.412·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.246535709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246535709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−15.36295345963924, −14.71586689634034, −14.38977632387740, −13.62792007507391, −13.06964033286086, −12.42474222851683, −12.19038720729400, −11.49990262730454, −10.99036138950962, −10.66715808069577, −9.589849689209853, −9.335117815750562, −8.982564282810996, −8.074463954836679, −7.263190290916904, −6.837047692659190, −6.571491227131236, −5.529904296707165, −5.288726034479435, −4.283774727233375, −3.653154746913398, −3.398230928629221, −2.165907264446104, −1.374048819399835, −0.4798969084839306,
0.4798969084839306, 1.374048819399835, 2.165907264446104, 3.398230928629221, 3.653154746913398, 4.283774727233375, 5.288726034479435, 5.529904296707165, 6.571491227131236, 6.837047692659190, 7.263190290916904, 8.074463954836679, 8.982564282810996, 9.335117815750562, 9.589849689209853, 10.66715808069577, 10.99036138950962, 11.49990262730454, 12.19038720729400, 12.42474222851683, 13.06964033286086, 13.62792007507391, 14.38977632387740, 14.71586689634034, 15.36295345963924
