| L(s) = 1 | + 5-s + 4·7-s + 13-s − 17-s + 4·19-s + 25-s − 6·29-s + 4·31-s + 4·35-s + 2·37-s − 6·41-s + 4·43-s + 9·49-s − 6·53-s − 12·59-s − 10·61-s + 65-s + 4·67-s − 12·71-s − 10·73-s + 16·79-s − 12·83-s − 85-s + 6·89-s + 4·91-s + 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.124·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s + 1.80·79-s − 1.31·83-s − 0.108·85-s + 0.635·89-s + 0.419·91-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 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
−13.66015843916403, −13.07079772033898, −12.62822524560523, −11.90368096124140, −11.63571923014358, −11.20757396567837, −10.64512727539553, −10.34208959618191, −9.635047647065809, −9.100635372692468, −8.851288466651731, −8.053070064032631, −7.770393817747961, −7.387010895524713, −6.594019993405109, −6.171546403004765, −5.422091750502935, −5.250840803140705, −4.475086364479152, −4.231158027782347, −3.301263227887928, −2.836160553428135, −2.018606309755306, −1.545438835152806, −1.079848734708199, 0,
1.079848734708199, 1.545438835152806, 2.018606309755306, 2.836160553428135, 3.301263227887928, 4.231158027782347, 4.475086364479152, 5.250840803140705, 5.422091750502935, 6.171546403004765, 6.594019993405109, 7.387010895524713, 7.770393817747961, 8.053070064032631, 8.851288466651731, 9.100635372692468, 9.635047647065809, 10.34208959618191, 10.64512727539553, 11.20757396567837, 11.63571923014358, 11.90368096124140, 12.62822524560523, 13.07079772033898, 13.66015843916403
