| L(s) = 1 | − 2.58·3-s − 3.96·5-s + 2.03·7-s + 3.69·9-s + 3.34·11-s + 3.37·13-s + 10.2·15-s + 6.55·17-s + 19-s − 5.26·21-s + 6.84·23-s + 10.7·25-s − 1.79·27-s + 3.95·29-s − 1.10·31-s − 8.66·33-s − 8.06·35-s + 1.25·37-s − 8.73·39-s + 4.56·41-s + 11.0·43-s − 14.6·45-s + 1.48·47-s − 2.85·49-s − 16.9·51-s + 0.297·53-s − 13.2·55-s + ⋯ |
| L(s) = 1 | − 1.49·3-s − 1.77·5-s + 0.769·7-s + 1.23·9-s + 1.00·11-s + 0.935·13-s + 2.64·15-s + 1.58·17-s + 0.229·19-s − 1.14·21-s + 1.42·23-s + 2.14·25-s − 0.346·27-s + 0.734·29-s − 0.199·31-s − 1.50·33-s − 1.36·35-s + 0.206·37-s − 1.39·39-s + 0.713·41-s + 1.67·43-s − 2.18·45-s + 0.216·47-s − 0.407·49-s − 2.37·51-s + 0.0408·53-s − 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239921281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239921281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 - 2.03T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 23 | \( 1 - 6.84T + 23T^{2} \) |
| 29 | \( 1 - 3.95T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 - 1.25T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 - 0.297T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 0.742T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−7.964931740642567224074228559868, −7.29447605945774783275862377877, −6.73540915632494746200643962819, −5.86864791216935824030872523439, −5.20049301149422282618924637139, −4.45570192527182179342888549002, −3.89419224262756257639805744335, −3.08403603424693006624194738182, −1.13866616565923983792344266837, −0.838677871500945668913012812331,
0.838677871500945668913012812331, 1.13866616565923983792344266837, 3.08403603424693006624194738182, 3.89419224262756257639805744335, 4.45570192527182179342888549002, 5.20049301149422282618924637139, 5.86864791216935824030872523439, 6.73540915632494746200643962819, 7.29447605945774783275862377877, 7.964931740642567224074228559868
