| L(s) = 1 | − 2.43·3-s − 0.700·5-s + 2.92·9-s − 6.25·11-s + 3.33·13-s + 1.70·15-s − 17-s + 2.78·19-s − 5.97·23-s − 4.50·25-s + 0.194·27-s − 3.96·29-s + 1.55·31-s + 15.2·33-s − 2.53·37-s − 8.12·39-s + 5.62·41-s − 2.29·43-s − 2.04·45-s − 11.9·47-s + 2.43·51-s + 6.55·53-s + 4.37·55-s − 6.77·57-s − 8.78·59-s + 4.95·61-s − 2.33·65-s + ⋯ |
| L(s) = 1 | − 1.40·3-s − 0.313·5-s + 0.973·9-s − 1.88·11-s + 0.925·13-s + 0.440·15-s − 0.242·17-s + 0.638·19-s − 1.24·23-s − 0.901·25-s + 0.0373·27-s − 0.736·29-s + 0.279·31-s + 2.64·33-s − 0.416·37-s − 1.30·39-s + 0.877·41-s − 0.349·43-s − 0.304·45-s − 1.74·47-s + 0.340·51-s + 0.900·53-s + 0.590·55-s − 0.897·57-s − 1.14·59-s + 0.634·61-s − 0.290·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4917780858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4917780858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 + 0.700T + 5T^{2} \) |
| 11 | \( 1 + 6.25T + 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 + 8.78T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 8.40T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 0.898T + 97T^{2} \) |
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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
−8.379476062494237666420072985023, −7.87040066890400722857062112066, −7.09964646898825519975260089062, −6.10921585531970728125078072490, −5.66815811017413852373645383493, −5.00227307683143456615607132168, −4.13525792869028015852812726466, −3.10503167744979575370749266539, −1.86967287483556514081064458890, −0.43475734108515463502525547193,
0.43475734108515463502525547193, 1.86967287483556514081064458890, 3.10503167744979575370749266539, 4.13525792869028015852812726466, 5.00227307683143456615607132168, 5.66815811017413852373645383493, 6.10921585531970728125078072490, 7.09964646898825519975260089062, 7.87040066890400722857062112066, 8.379476062494237666420072985023
