L(s) = 1 | + 2·5-s + 7-s + 3·11-s − 2·17-s − 2·19-s − 4·23-s − 25-s + 2·29-s + 10·31-s + 2·35-s + 9·37-s − 2·41-s − 43-s + 8·47-s + 49-s + 11·53-s + 6·55-s + 6·59-s − 8·61-s − 3·67-s + 3·71-s + 4·73-s + 3·77-s + 79-s + 6·83-s − 4·85-s + 4·89-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.377·7-s + 0.904·11-s − 0.485·17-s − 0.458·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 1.79·31-s + 0.338·35-s + 1.47·37-s − 0.312·41-s − 0.152·43-s + 1.16·47-s + 1/7·49-s + 1.51·53-s + 0.809·55-s + 0.781·59-s − 1.02·61-s − 0.366·67-s + 0.356·71-s + 0.468·73-s + 0.341·77-s + 0.112·79-s + 0.658·83-s − 0.433·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.606204314\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606204314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.402773602693818822026247229596, −7.67188577237697185267550322771, −6.65570018730583887555357322639, −6.22662054902984970122706181668, −5.50906955083684714686429525487, −4.50449775435939553268652534010, −3.97554023181141409715836466376, −2.67970398699750458834376931624, −1.97851182872562845647314172121, −0.938284903546578701335889112674,
0.938284903546578701335889112674, 1.97851182872562845647314172121, 2.67970398699750458834376931624, 3.97554023181141409715836466376, 4.50449775435939553268652534010, 5.50906955083684714686429525487, 6.22662054902984970122706181668, 6.65570018730583887555357322639, 7.67188577237697185267550322771, 8.402773602693818822026247229596