| L(s) = 1 | + 3.23·3-s + 3.17·5-s + 7.48·9-s + 2.78·11-s − 5.61·13-s + 10.2·15-s + 2.92·17-s + 19-s + 6.95·23-s + 5.10·25-s + 14.5·27-s − 4.88·29-s − 4.91·31-s + 9.00·33-s + 1.15·37-s − 18.1·39-s − 4.32·41-s − 10.6·43-s + 23.7·45-s − 9.64·47-s + 9.45·51-s − 6.26·53-s + 8.83·55-s + 3.23·57-s − 5.91·59-s + 6.96·61-s − 17.8·65-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 1.42·5-s + 2.49·9-s + 0.838·11-s − 1.55·13-s + 2.65·15-s + 0.708·17-s + 0.229·19-s + 1.45·23-s + 1.02·25-s + 2.79·27-s − 0.906·29-s − 0.883·31-s + 1.56·33-s + 0.190·37-s − 2.90·39-s − 0.674·41-s − 1.62·43-s + 3.54·45-s − 1.40·47-s + 1.32·51-s − 0.861·53-s + 1.19·55-s + 0.428·57-s − 0.769·59-s + 0.891·61-s − 2.21·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.238983453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.238983453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 + 4.91T + 31T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 + 4.32T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.64T + 47T^{2} \) |
| 53 | \( 1 + 6.26T + 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 - 0.156T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 3.48T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.715006386899096462249818364185, −7.82887823109003117612036425068, −7.13569964636920035186747439810, −6.58623164598755074111940001278, −5.35442409350998846362627690028, −4.72395929262658246535685385918, −3.52273313709299378174999463999, −2.95048219673612103731649088490, −2.02604116217972944272622510662, −1.47810733044619875012923818120,
1.47810733044619875012923818120, 2.02604116217972944272622510662, 2.95048219673612103731649088490, 3.52273313709299378174999463999, 4.72395929262658246535685385918, 5.35442409350998846362627690028, 6.58623164598755074111940001278, 7.13569964636920035186747439810, 7.82887823109003117612036425068, 8.715006386899096462249818364185
