| L(s) = 1 | + 3-s − 2.64·5-s + 7-s + 9-s + 6.24·11-s + 4·13-s − 2.64·15-s − 2.64·17-s + 19-s + 21-s + 6.24·23-s + 1.96·25-s + 27-s + 3.60·29-s + 8.24·31-s + 6.24·33-s − 2.64·35-s + 2·37-s + 4·39-s − 5.21·41-s − 4.24·43-s − 2.64·45-s − 1.60·47-s + 49-s − 2.64·51-s − 12.8·53-s − 16.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.18·5-s + 0.377·7-s + 0.333·9-s + 1.88·11-s + 1.10·13-s − 0.681·15-s − 0.640·17-s + 0.229·19-s + 0.218·21-s + 1.30·23-s + 0.393·25-s + 0.192·27-s + 0.670·29-s + 1.48·31-s + 1.08·33-s − 0.446·35-s + 0.328·37-s + 0.640·39-s − 0.815·41-s − 0.648·43-s − 0.393·45-s − 0.234·47-s + 0.142·49-s − 0.369·51-s − 1.77·53-s − 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.675379034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675379034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 - 0.0605T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 0.329T + 71T^{2} \) |
| 73 | \( 1 + 1.21T + 73T^{2} \) |
| 79 | \( 1 + 0.969T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 - 9.03T + 97T^{2} \) |
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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
−8.163556567202785062837310357135, −7.38463198361793927193907405305, −6.63080931970794129547440742368, −6.22630080984453202101342955366, −4.78749299024637643212356796634, −4.36110518447050561736124860025, −3.57399395075995681536588629263, −3.06538824690651378290260032082, −1.65936719572374694223231835700, −0.904788374363327298438206443996,
0.904788374363327298438206443996, 1.65936719572374694223231835700, 3.06538824690651378290260032082, 3.57399395075995681536588629263, 4.36110518447050561736124860025, 4.78749299024637643212356796634, 6.22630080984453202101342955366, 6.63080931970794129547440742368, 7.38463198361793927193907405305, 8.163556567202785062837310357135
