| L(s) = 1 | − 3-s − 3.43·7-s + 9-s − 2.63·11-s + 13-s − 7.84·17-s + 0.794·19-s + 3.43·21-s + 3.43·23-s − 27-s − 4.06·29-s + 9.27·31-s + 2.63·33-s + 0.636·37-s − 39-s − 4.63·41-s − 2.41·43-s − 5.27·47-s + 4.77·49-s + 7.84·51-s − 11.4·53-s − 0.794·57-s − 12.1·59-s − 11.4·61-s − 3.43·63-s + 6.41·67-s − 3.43·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.29·7-s + 0.333·9-s − 0.795·11-s + 0.277·13-s − 1.90·17-s + 0.182·19-s + 0.748·21-s + 0.715·23-s − 0.192·27-s − 0.755·29-s + 1.66·31-s + 0.458·33-s + 0.104·37-s − 0.160·39-s − 0.724·41-s − 0.367·43-s − 0.769·47-s + 0.682·49-s + 1.09·51-s − 1.57·53-s − 0.105·57-s − 1.57·59-s − 1.47·61-s − 0.432·63-s + 0.783·67-s − 0.413·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5648577413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5648577413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 17 | \( 1 + 7.84T + 17T^{2} \) |
| 19 | \( 1 - 0.794T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 - 0.636T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.75625601805080866672477189897, −6.99377539923101745189646758395, −6.33926956923891183443138056979, −6.06637924896741067587609129200, −4.90291986094831428099672982859, −4.53937453136640516489088813889, −3.38732111244310785176701667101, −2.82491329707701183641655208417, −1.75692941219339960756662673265, −0.37001951311171439947614146040,
0.37001951311171439947614146040, 1.75692941219339960756662673265, 2.82491329707701183641655208417, 3.38732111244310785176701667101, 4.53937453136640516489088813889, 4.90291986094831428099672982859, 6.06637924896741067587609129200, 6.33926956923891183443138056979, 6.99377539923101745189646758395, 7.75625601805080866672477189897
