| L(s) = 1 | + 0.940·2-s + 3-s − 1.11·4-s + 5-s + 0.940·6-s + 0.900·7-s − 2.92·8-s + 9-s + 0.940·10-s + 4.51·11-s − 1.11·12-s − 6.09·13-s + 0.847·14-s + 15-s − 0.523·16-s − 1.27·17-s + 0.940·18-s + 1.58·19-s − 1.11·20-s + 0.900·21-s + 4.24·22-s − 2.92·24-s + 25-s − 5.73·26-s + 27-s − 1.00·28-s + 3.74·29-s + ⋯ |
| L(s) = 1 | + 0.664·2-s + 0.577·3-s − 0.557·4-s + 0.447·5-s + 0.383·6-s + 0.340·7-s − 1.03·8-s + 0.333·9-s + 0.297·10-s + 1.36·11-s − 0.322·12-s − 1.69·13-s + 0.226·14-s + 0.258·15-s − 0.130·16-s − 0.308·17-s + 0.221·18-s + 0.363·19-s − 0.249·20-s + 0.196·21-s + 0.904·22-s − 0.598·24-s + 0.200·25-s − 1.12·26-s + 0.192·27-s − 0.189·28-s + 0.695·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.338255576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.338255576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.940T + 2T^{2} \) |
| 7 | \( 1 - 0.900T + 7T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 6.80T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 + 7.87T + 89T^{2} \) |
| 97 | \( 1 - 7.91T + 97T^{2} \) |
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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
−8.030673388408194150281383176261, −6.79501776122014245397737681524, −6.68824836827108406594936932228, −5.50204068175433383418674290797, −4.94086983632289197456394994123, −4.34756631169818530847692821468, −3.61424527652925641029530282986, −2.77684891776087801213363635140, −1.99678765586516316258776368934, −0.807219643272305214844210269750,
0.807219643272305214844210269750, 1.99678765586516316258776368934, 2.77684891776087801213363635140, 3.61424527652925641029530282986, 4.34756631169818530847692821468, 4.94086983632289197456394994123, 5.50204068175433383418674290797, 6.68824836827108406594936932228, 6.79501776122014245397737681524, 8.030673388408194150281383176261
