| L(s) = 1 | − 3-s + 2i·5-s + 4i·7-s + 9-s − 2i·15-s − 2·17-s − 8i·19-s − 4i·21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4i·31-s − 8·35-s − 10i·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894i·5-s + 1.51i·7-s + 0.333·9-s − 0.516i·15-s − 0.485·17-s − 1.83i·19-s − 0.872i·21-s − 1.66·23-s + 0.200·25-s − 0.192·27-s − 0.371·29-s − 0.718i·31-s − 1.35·35-s − 1.64i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7477560538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7477560538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 14iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.469406958442807821428975525692, −7.35016765942482375773154320930, −6.86340162247837341881340829021, −5.96440628541283493888670961192, −5.59789882671124399712988560726, −4.63953879852879865856699751452, −3.69947440476043309253858391302, −2.50720558585888979910815343873, −2.17603307483345049820413325202, −0.26360063506803633428966836616,
1.00412164802975041055540353189, 1.72434240454804191123836566322, 3.35156696219889861450894455035, 4.25091156028158522063167245238, 4.56949441544393243868015101293, 5.64833471268085440419633784565, 6.27214961142902803845116249639, 7.09998081692499823636777263886, 7.898380856199665783296892544218, 8.320369693221325341027655802851
