| L(s) = 1 | + 3·3-s − 3·5-s + 2·7-s + 6·9-s + 11-s + 3·13-s − 9·15-s − 4·17-s + 8·19-s + 6·21-s + 4·25-s + 9·27-s − 29-s − 3·31-s + 3·33-s − 6·35-s − 8·37-s + 9·39-s − 2·41-s − 7·43-s − 18·45-s − 11·47-s − 3·49-s − 12·51-s + 53-s − 3·55-s + 24·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s + 0.301·11-s + 0.832·13-s − 2.32·15-s − 0.970·17-s + 1.83·19-s + 1.30·21-s + 4/5·25-s + 1.73·27-s − 0.185·29-s − 0.538·31-s + 0.522·33-s − 1.01·35-s − 1.31·37-s + 1.44·39-s − 0.312·41-s − 1.06·43-s − 2.68·45-s − 1.60·47-s − 3/7·49-s − 1.68·51-s + 0.137·53-s − 0.404·55-s + 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.223609629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223609629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.23300559069210877573679853901, −9.934847997724423333490222503104, −8.900780690155322256875750624250, −8.356447790872927004036634139026, −7.66665834440110862868536116407, −6.89961106208360374634086167814, −4.94755986718525722442660740269, −3.83674514665941575921202847704, −3.24462917810006223656521762150, −1.65158165921267727855028362620,
1.65158165921267727855028362620, 3.24462917810006223656521762150, 3.83674514665941575921202847704, 4.94755986718525722442660740269, 6.89961106208360374634086167814, 7.66665834440110862868536116407, 8.356447790872927004036634139026, 8.900780690155322256875750624250, 9.934847997724423333490222503104, 11.23300559069210877573679853901
