| L(s) = 1 | − 0.686·2-s − 1.52·4-s − 3.77·5-s + 4.48·7-s + 2.42·8-s + 2.59·10-s − 11-s − 0.769·13-s − 3.07·14-s + 1.39·16-s + 3.80·17-s + 6.11·19-s + 5.77·20-s + 0.686·22-s + 3.48·23-s + 9.26·25-s + 0.528·26-s − 6.85·28-s − 4.50·29-s + 1.54·31-s − 5.80·32-s − 2.60·34-s − 16.9·35-s + 10.3·37-s − 4.19·38-s − 9.14·40-s + 1.01·41-s + ⋯ |
| L(s) = 1 | − 0.485·2-s − 0.764·4-s − 1.68·5-s + 1.69·7-s + 0.856·8-s + 0.819·10-s − 0.301·11-s − 0.213·13-s − 0.821·14-s + 0.349·16-s + 0.921·17-s + 1.40·19-s + 1.29·20-s + 0.146·22-s + 0.726·23-s + 1.85·25-s + 0.103·26-s − 1.29·28-s − 0.837·29-s + 0.277·31-s − 1.02·32-s − 0.447·34-s − 2.86·35-s + 1.69·37-s − 0.680·38-s − 1.44·40-s + 0.157·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135623526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135623526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.686T + 2T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 4.48T + 7T^{2} \) |
| 13 | \( 1 + 0.769T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 67 | \( 1 + 7.87T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.966843351525080390772556515778, −7.60476565064891766305168921830, −7.28114758935959438406173523102, −5.66722890074030495866590340871, −4.98739528552554785033856065802, −4.50426985894595816213136529184, −3.80449553343394616084647251920, −2.92064773240923451334424877512, −1.41060211886554269871241498241, −0.69009158803648905573931301033,
0.69009158803648905573931301033, 1.41060211886554269871241498241, 2.92064773240923451334424877512, 3.80449553343394616084647251920, 4.50426985894595816213136529184, 4.98739528552554785033856065802, 5.66722890074030495866590340871, 7.28114758935959438406173523102, 7.60476565064891766305168921830, 7.966843351525080390772556515778
