| L(s) = 1 | + 3.00·3-s + 1.81·5-s + 6.00·9-s + 3.65·11-s + 1.31·13-s + 5.45·15-s − 1.00·17-s − 19-s − 2.54·23-s − 1.69·25-s + 9.01·27-s − 1.89·29-s + 7.63·31-s + 10.9·33-s + 5.65·37-s + 3.93·39-s + 3.71·41-s + 2.53·43-s + 10.9·45-s − 7.38·47-s − 3.00·51-s − 9.27·53-s + 6.64·55-s − 3.00·57-s + 5.58·59-s − 6.40·61-s + 2.38·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.813·5-s + 2.00·9-s + 1.10·11-s + 0.364·13-s + 1.40·15-s − 0.243·17-s − 0.229·19-s − 0.531·23-s − 0.338·25-s + 1.73·27-s − 0.351·29-s + 1.37·31-s + 1.90·33-s + 0.930·37-s + 0.630·39-s + 0.580·41-s + 0.386·43-s + 1.62·45-s − 1.07·47-s − 0.421·51-s − 1.27·53-s + 0.896·55-s − 0.397·57-s + 0.727·59-s − 0.820·61-s + 0.296·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.403377877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.403377877\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 - 8.36T + 67T^{2} \) |
| 71 | \( 1 - 0.250T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.060720004046944851864413865760, −7.38154902660913354399981846374, −6.43213919748530440259125062504, −6.12183527448315459791798170156, −4.85416726553570369746630587408, −4.08085925350834603223246323940, −3.51427809937714908547730572744, −2.59989985038651079397634452892, −1.97973644690762720343901329871, −1.18297985817660098186449556162,
1.18297985817660098186449556162, 1.97973644690762720343901329871, 2.59989985038651079397634452892, 3.51427809937714908547730572744, 4.08085925350834603223246323940, 4.85416726553570369746630587408, 6.12183527448315459791798170156, 6.43213919748530440259125062504, 7.38154902660913354399981846374, 8.060720004046944851864413865760
