| L(s) = 1 | + 2-s + 3-s + 4-s + 3.70·5-s + 6-s + 8-s + 9-s + 3.70·10-s + 5.70·11-s + 12-s − 13-s + 3.70·15-s + 16-s − 3.70·17-s + 18-s − 5.70·19-s + 3.70·20-s + 5.70·22-s − 1.70·23-s + 24-s + 8.70·25-s − 26-s + 27-s − 3.70·29-s + 3.70·30-s + 32-s + 5.70·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.17·10-s + 1.71·11-s + 0.288·12-s − 0.277·13-s + 0.955·15-s + 0.250·16-s − 0.897·17-s + 0.235·18-s − 1.30·19-s + 0.827·20-s + 1.21·22-s − 0.354·23-s + 0.204·24-s + 1.74·25-s − 0.196·26-s + 0.192·27-s − 0.687·29-s + 0.675·30-s + 0.176·32-s + 0.992·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.500330280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.500330280\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.70T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 + 3.40T + 79T^{2} \) |
| 83 | \( 1 - 0.596T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.792089130817399362758465445364, −7.59584046041812661538650921092, −6.75608033364335078939690047310, −6.16744775757243342477793843212, −5.74292246227952023492002227871, −4.42456190823828619421873725895, −4.12674046533058564668665622287, −2.80037503134609174492594909662, −2.12843806786283010744424863656, −1.38638969543504239388096097368,
1.38638969543504239388096097368, 2.12843806786283010744424863656, 2.80037503134609174492594909662, 4.12674046533058564668665622287, 4.42456190823828619421873725895, 5.74292246227952023492002227871, 6.16744775757243342477793843212, 6.75608033364335078939690047310, 7.59584046041812661538650921092, 8.792089130817399362758465445364
