| L(s) = 1 | + 3-s + 5-s + 9-s + 13-s + 15-s − 7·17-s + 5·19-s + 2·23-s + 25-s + 27-s − 4·29-s − 2·31-s − 7·37-s + 39-s + 9·41-s + 11·43-s + 45-s − 3·47-s − 7·49-s − 7·51-s + 5·57-s + 2·59-s + 14·61-s + 65-s − 7·67-s + 2·69-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 1.69·17-s + 1.14·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.359·31-s − 1.15·37-s + 0.160·39-s + 1.40·41-s + 1.67·43-s + 0.149·45-s − 0.437·47-s − 49-s − 0.980·51-s + 0.662·57-s + 0.260·59-s + 1.79·61-s + 0.124·65-s − 0.855·67-s + 0.240·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.309285676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.309285676\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−12.96481961661779, −12.88319143209009, −12.38403297027313, −11.46530372470363, −11.28654074732022, −10.83780443103956, −10.24956710655672, −9.607283319558676, −9.367528706335516, −8.889569205608578, −8.445880167781516, −7.894173278354782, −7.254622341639004, −6.947228446310140, −6.429310052251314, −5.711921920736721, −5.366497656085031, −4.712605485513625, −4.103644113389500, −3.702419808154658, −2.934725537328773, −2.506254057419001, −1.894991476287648, −1.304483358838022, −0.4943762996062632,
0.4943762996062632, 1.304483358838022, 1.894991476287648, 2.506254057419001, 2.934725537328773, 3.702419808154658, 4.103644113389500, 4.712605485513625, 5.366497656085031, 5.711921920736721, 6.429310052251314, 6.947228446310140, 7.254622341639004, 7.894173278354782, 8.445880167781516, 8.889569205608578, 9.367528706335516, 9.607283319558676, 10.24956710655672, 10.83780443103956, 11.28654074732022, 11.46530372470363, 12.38403297027313, 12.88319143209009, 12.96481961661779
