| L(s) = 1 | + 1.26·2-s + 1.44·3-s − 0.390·4-s − 3.92·5-s + 1.83·6-s + 0.416·7-s − 3.03·8-s − 0.900·9-s − 4.98·10-s + 2.26·11-s − 0.565·12-s − 5.30·13-s + 0.527·14-s − 5.69·15-s − 3.06·16-s + 17-s − 1.14·18-s − 5.97·19-s + 1.53·20-s + 0.602·21-s + 2.87·22-s + 0.261·23-s − 4.39·24-s + 10.4·25-s − 6.72·26-s − 5.65·27-s − 0.162·28-s + ⋯ |
| L(s) = 1 | + 0.897·2-s + 0.836·3-s − 0.195·4-s − 1.75·5-s + 0.750·6-s + 0.157·7-s − 1.07·8-s − 0.300·9-s − 1.57·10-s + 0.682·11-s − 0.163·12-s − 1.47·13-s + 0.141·14-s − 1.47·15-s − 0.766·16-s + 0.242·17-s − 0.269·18-s − 1.37·19-s + 0.342·20-s + 0.131·21-s + 0.612·22-s + 0.0545·23-s − 0.896·24-s + 2.08·25-s − 1.31·26-s − 1.08·27-s − 0.0306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 0.416T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 - 0.261T + 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 + 5.44T + 37T^{2} \) |
| 41 | \( 1 - 0.154T + 41T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 1.25T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 + 1.33T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.826988664607759185826003017771, −8.825788711114807019967575308260, −8.291416258369184164754109138405, −7.47750776604996677130551725674, −6.44485805639373041636308538271, −5.01616787705797303775654692140, −4.26616865473759799590223774586, −3.55288220085207534472999399257, −2.61329461351816345442309042313, 0,
2.61329461351816345442309042313, 3.55288220085207534472999399257, 4.26616865473759799590223774586, 5.01616787705797303775654692140, 6.44485805639373041636308538271, 7.47750776604996677130551725674, 8.291416258369184164754109138405, 8.825788711114807019967575308260, 9.826988664607759185826003017771
