| L(s) = 1 | − 3-s + 5-s − 2·9-s + 3·11-s + 6·13-s − 15-s − 5·17-s + 19-s + 7·23-s − 4·25-s + 5·27-s − 2·29-s + 5·31-s − 3·33-s − 3·37-s − 6·39-s − 2·41-s − 4·43-s − 2·45-s − 5·47-s + 5·51-s + 53-s + 3·55-s − 57-s + 15·59-s + 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s + 1.66·13-s − 0.258·15-s − 1.21·17-s + 0.229·19-s + 1.45·23-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.898·31-s − 0.522·33-s − 0.493·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.729·47-s + 0.700·51-s + 0.137·53-s + 0.404·55-s − 0.132·57-s + 1.95·59-s + 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.687744586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.687744586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.806052835295510180982841265322, −8.122400564979183537512281772546, −6.78488200573146388250216630872, −6.48888869314089673012158829520, −5.72399450043503792345692082266, −4.97886361799519777103739823140, −3.97021165488683054609185324464, −3.14587189012527864759158099859, −1.91283106492490953611536926560, −0.833715032706912072132184635317,
0.833715032706912072132184635317, 1.91283106492490953611536926560, 3.14587189012527864759158099859, 3.97021165488683054609185324464, 4.97886361799519777103739823140, 5.72399450043503792345692082266, 6.48888869314089673012158829520, 6.78488200573146388250216630872, 8.122400564979183537512281772546, 8.806052835295510180982841265322
