| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 4·13-s + 15-s + 6·17-s − 8·19-s + 2·21-s + 25-s − 27-s − 31-s + 2·35-s − 4·37-s + 4·39-s − 6·41-s − 8·43-s − 45-s + 12·47-s − 3·49-s − 6·51-s − 6·53-s + 8·57-s + 6·59-s + 2·61-s − 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.179·31-s + 0.338·35-s − 0.657·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 1.05·57-s + 0.781·59-s + 0.256·61-s − 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6395205900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6395205900\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.87639961137077448711666218442, −6.99995037654568993530287089242, −6.66487513131472998169095616295, −5.75648357361685279960819834335, −5.15112698677463794171562445987, −4.33514787626531739002349890306, −3.60390488386775971089345612863, −2.77796430070645379020467842410, −1.73954738745817360051187806680, −0.40261550052433971362564040935,
0.40261550052433971362564040935, 1.73954738745817360051187806680, 2.77796430070645379020467842410, 3.60390488386775971089345612863, 4.33514787626531739002349890306, 5.15112698677463794171562445987, 5.75648357361685279960819834335, 6.66487513131472998169095616295, 6.99995037654568993530287089242, 7.87639961137077448711666218442
