| L(s) = 1 | + 2.61·2-s + 4.85·4-s + 3·5-s − 3.85·7-s + 7.47·8-s + 7.85·10-s − 2.23·11-s − 0.145·13-s − 10.0·14-s + 9.85·16-s + 0.763·17-s − 2.85·19-s + 14.5·20-s − 5.85·22-s − 7.47·23-s + 4·25-s − 0.381·26-s − 18.7·28-s + 7.47·29-s − 31-s + 10.8·32-s + 2·34-s − 11.5·35-s − 3.85·37-s − 7.47·38-s + 22.4·40-s + 0.381·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s + 1.34·5-s − 1.45·7-s + 2.64·8-s + 2.48·10-s − 0.674·11-s − 0.0404·13-s − 2.69·14-s + 2.46·16-s + 0.185·17-s − 0.654·19-s + 3.25·20-s − 1.24·22-s − 1.55·23-s + 0.800·25-s − 0.0749·26-s − 3.53·28-s + 1.38·29-s − 0.179·31-s + 1.91·32-s + 0.342·34-s − 1.95·35-s − 0.633·37-s − 1.21·38-s + 3.54·40-s + 0.0596·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.461993876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461993876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.145T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 3.85T + 37T^{2} \) |
| 41 | \( 1 - 0.381T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−10.47682980311921097606303870627, −10.28624093620410668705708640710, −9.113248344626360654061989817746, −7.54800025374035698216042337385, −6.34669456238856345921073125474, −6.14177312502520564955591777175, −5.22791143322054589825114302638, −4.05691158566673723646147874888, −2.93773178686856318444319071104, −2.15011980704806818069036716496,
2.15011980704806818069036716496, 2.93773178686856318444319071104, 4.05691158566673723646147874888, 5.22791143322054589825114302638, 6.14177312502520564955591777175, 6.34669456238856345921073125474, 7.54800025374035698216042337385, 9.113248344626360654061989817746, 10.28624093620410668705708640710, 10.47682980311921097606303870627
