| L(s) = 1 | − 0.319·2-s + 0.543·3-s − 1.89·4-s + 1.90·5-s − 0.173·6-s + 0.747·7-s + 1.24·8-s − 2.70·9-s − 0.609·10-s − 5.00·11-s − 1.03·12-s − 13-s − 0.238·14-s + 1.03·15-s + 3.39·16-s + 1.16·17-s + 0.863·18-s + 7.09·19-s − 3.61·20-s + 0.406·21-s + 1.59·22-s + 7.48·23-s + 0.677·24-s − 1.36·25-s + 0.319·26-s − 3.10·27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.225·2-s + 0.314·3-s − 0.949·4-s + 0.852·5-s − 0.0709·6-s + 0.282·7-s + 0.440·8-s − 0.901·9-s − 0.192·10-s − 1.50·11-s − 0.298·12-s − 0.277·13-s − 0.0637·14-s + 0.267·15-s + 0.849·16-s + 0.283·17-s + 0.203·18-s + 1.62·19-s − 0.809·20-s + 0.0886·21-s + 0.340·22-s + 1.56·23-s + 0.138·24-s − 0.272·25-s + 0.0626·26-s − 0.597·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398199255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398199255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.319T + 2T^{2} \) |
| 3 | \( 1 - 0.543T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 - 0.747T + 7T^{2} \) |
| 11 | \( 1 + 5.00T + 11T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 9.53T + 29T^{2} \) |
| 31 | \( 1 - 1.40T + 31T^{2} \) |
| 37 | \( 1 - 1.98T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 + 7.21T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 - 4.65T + 67T^{2} \) |
| 71 | \( 1 + 9.57T + 71T^{2} \) |
| 73 | \( 1 - 6.07T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 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
−9.520476746989406462533368935312, −8.932789290177837325035345040439, −8.006074681489811848793645857768, −7.57020976919102025849631039326, −6.11055931203371896730907151387, −5.21228068260362214840957825774, −4.88808999799157684339362232838, −3.26271858282884225594914283880, −2.52278654867331470383348882753, −0.899866840271059074571982930379,
0.899866840271059074571982930379, 2.52278654867331470383348882753, 3.26271858282884225594914283880, 4.88808999799157684339362232838, 5.21228068260362214840957825774, 6.11055931203371896730907151387, 7.57020976919102025849631039326, 8.006074681489811848793645857768, 8.932789290177837325035345040439, 9.520476746989406462533368935312
