| L(s) = 1 | − 3-s − 1.63·5-s + 0.404·7-s + 9-s − 2.06·11-s − 1.45·13-s + 1.63·15-s + 0.774·17-s + 4.43·19-s − 0.404·21-s − 5.99·23-s − 2.33·25-s − 27-s + 8.42·29-s + 1.79·31-s + 2.06·33-s − 0.660·35-s − 4.57·37-s + 1.45·39-s + 12.4·41-s + 6.03·43-s − 1.63·45-s + 7.80·47-s − 6.83·49-s − 0.774·51-s − 5.39·53-s + 3.36·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.730·5-s + 0.152·7-s + 0.333·9-s − 0.621·11-s − 0.402·13-s + 0.421·15-s + 0.187·17-s + 1.01·19-s − 0.0882·21-s − 1.25·23-s − 0.466·25-s − 0.192·27-s + 1.56·29-s + 0.321·31-s + 0.358·33-s − 0.111·35-s − 0.751·37-s + 0.232·39-s + 1.95·41-s + 0.920·43-s − 0.243·45-s + 1.13·47-s − 0.976·49-s − 0.108·51-s − 0.741·53-s + 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 0.404T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 - 0.774T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 + 5.39T + 53T^{2} \) |
| 59 | \( 1 - 6.41T + 59T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 0.422T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−7.65825630287435718845289203983, −7.21758088621629199592916229670, −6.14652155741302143231596590076, −5.66495481537002156996755928459, −4.72582362793407357650911297775, −4.24155212972894794868849390588, −3.24966212554431976193989018953, −2.37548269055608191238043076118, −1.10901471719398478190242607592, 0,
1.10901471719398478190242607592, 2.37548269055608191238043076118, 3.24966212554431976193989018953, 4.24155212972894794868849390588, 4.72582362793407357650911297775, 5.66495481537002156996755928459, 6.14652155741302143231596590076, 7.21758088621629199592916229670, 7.65825630287435718845289203983
