| L(s) = 1 | + 0.692·2-s + 1.14·3-s − 1.52·4-s + 0.794·6-s + 3.60·7-s − 2.43·8-s − 1.68·9-s + 11-s − 1.74·12-s + 7.02·13-s + 2.49·14-s + 1.35·16-s + 0.385·17-s − 1.16·18-s + 19-s + 4.13·21-s + 0.692·22-s − 4.88·23-s − 2.79·24-s + 4.86·26-s − 5.37·27-s − 5.47·28-s − 3.20·29-s + 4.99·31-s + 5.81·32-s + 1.14·33-s + 0.266·34-s + ⋯ |
| L(s) = 1 | + 0.489·2-s + 0.662·3-s − 0.760·4-s + 0.324·6-s + 1.36·7-s − 0.862·8-s − 0.561·9-s + 0.301·11-s − 0.503·12-s + 1.94·13-s + 0.667·14-s + 0.337·16-s + 0.0934·17-s − 0.274·18-s + 0.229·19-s + 0.902·21-s + 0.147·22-s − 1.01·23-s − 0.570·24-s + 0.954·26-s − 1.03·27-s − 1.03·28-s − 0.595·29-s + 0.897·31-s + 1.02·32-s + 0.199·33-s + 0.0457·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189091199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189091199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.692T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 - 7.02T + 13T^{2} \) |
| 17 | \( 1 - 0.385T + 17T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 + 0.661T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 5.81T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.343832111299852949943767436305, −7.87544054148205700785118745716, −6.65458774230530276715959179365, −5.71408750151617445569497334272, −5.42659195118253433696985312746, −4.23401148534485417222225087947, −3.92114997519883480272423181582, −3.04308410958815643884425052549, −1.94576835492590916691205360207, −0.923161087003682367719686400778,
0.923161087003682367719686400778, 1.94576835492590916691205360207, 3.04308410958815643884425052549, 3.92114997519883480272423181582, 4.23401148534485417222225087947, 5.42659195118253433696985312746, 5.71408750151617445569497334272, 6.65458774230530276715959179365, 7.87544054148205700785118745716, 8.343832111299852949943767436305
