| L(s) = 1 | + 7-s − 3.46·11-s − 4·13-s − 6.92·17-s + 19-s − 3.46·23-s − 5·25-s + 3.46·29-s + 8·31-s + 2·37-s + 6.92·41-s + 8·43-s − 3.46·47-s + 49-s + 10.3·53-s + 13.8·59-s + 2·61-s + 2·67-s − 3.46·71-s + 14·73-s − 3.46·77-s + 14·79-s + 10.3·83-s − 13.8·89-s − 4·91-s + 8·97-s − 4·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.04·11-s − 1.10·13-s − 1.68·17-s + 0.229·19-s − 0.722·23-s − 25-s + 0.643·29-s + 1.43·31-s + 0.328·37-s + 1.08·41-s + 1.21·43-s − 0.505·47-s + 0.142·49-s + 1.42·53-s + 1.80·59-s + 0.256·61-s + 0.244·67-s − 0.411·71-s + 1.63·73-s − 0.394·77-s + 1.57·79-s + 1.14·83-s − 1.46·89-s − 0.419·91-s + 0.812·97-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.395034155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395034155\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.146126621577866774811944768055, −7.68605915912554346282636713151, −6.88363124973409026553181559355, −6.11596017977960070102375215482, −5.25436758350431740350857426748, −4.61315100650871195355261603975, −3.92025909385226681172064085787, −2.44080547567522048938679262169, −2.33846848866104011986554243357, −0.61990338528487705978051009426,
0.61990338528487705978051009426, 2.33846848866104011986554243357, 2.44080547567522048938679262169, 3.92025909385226681172064085787, 4.61315100650871195355261603975, 5.25436758350431740350857426748, 6.11596017977960070102375215482, 6.88363124973409026553181559355, 7.68605915912554346282636713151, 8.146126621577866774811944768055
