| L(s) = 1 | + 3-s + 4·5-s + 9-s − 4·13-s + 4·15-s − 17-s + 8·19-s + 11·25-s + 27-s + 8·29-s − 4·31-s + 11·37-s − 4·39-s − 9·41-s − 2·43-s + 4·45-s − 2·47-s − 7·49-s − 51-s − 3·53-s + 8·57-s − 9·59-s + 13·61-s − 16·65-s + 14·67-s + 5·71-s + 16·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.10·13-s + 1.03·15-s − 0.242·17-s + 1.83·19-s + 11/5·25-s + 0.192·27-s + 1.48·29-s − 0.718·31-s + 1.80·37-s − 0.640·39-s − 1.40·41-s − 0.304·43-s + 0.596·45-s − 0.291·47-s − 49-s − 0.140·51-s − 0.412·53-s + 1.05·57-s − 1.17·59-s + 1.66·61-s − 1.98·65-s + 1.71·67-s + 0.593·71-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.742953011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.742953011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.92428931611252, −13.31565203133664, −12.97536155588707, −12.40316446562745, −11.93854993503482, −11.23552989082165, −10.74512315789389, −9.943663678947247, −9.790610312208983, −9.508845817839744, −9.010452887045587, −8.173666591064162, −7.927792413646076, −7.075086650852332, −6.696637378501826, −6.224342206956580, −5.485529396295491, −4.938507675627019, −4.840589945222803, −3.671792695255930, −3.119290959956101, −2.489312963370453, −2.136969980746537, −1.374197174499013, −0.7412828028425318,
0.7412828028425318, 1.374197174499013, 2.136969980746537, 2.489312963370453, 3.119290959956101, 3.671792695255930, 4.840589945222803, 4.938507675627019, 5.485529396295491, 6.224342206956580, 6.696637378501826, 7.075086650852332, 7.927792413646076, 8.173666591064162, 9.010452887045587, 9.508845817839744, 9.790610312208983, 9.943663678947247, 10.74512315789389, 11.23552989082165, 11.93854993503482, 12.40316446562745, 12.97536155588707, 13.31565203133664, 13.92428931611252
