| L(s) = 1 | − i·2-s − 4-s − 2i·7-s + i·8-s − 11-s − 4i·13-s − 2·14-s + 16-s + 2i·17-s + i·22-s − 6i·23-s − 4·26-s + 2i·28-s + 10·29-s − 8·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s − 0.301·11-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s + 0.485i·17-s + 0.213i·22-s − 1.25i·23-s − 0.784·26-s + 0.377i·28-s + 1.85·29-s − 1.43·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8702324191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8702324191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.976653133544999287645231987049, −7.22585867775822784695178785376, −6.36911340751780402912321123510, −5.52938004927756755668258579858, −4.73421603678092641613555779296, −4.01452062216147973724739013473, −3.18055419404108722373996368765, −2.41957937002991134222393223788, −1.23088081893904090624068388133, −0.24979717567563290337473301274,
1.40726769819070323241315914388, 2.49496653543703707226291001882, 3.44768547244141371661170054413, 4.42074371927323870175155787073, 5.12208885493177743916003754334, 5.78204925255615844704764562763, 6.54594956088119559614825036239, 7.18759578389728482940487450710, 7.87742752029273934567738656972, 8.715936002853385954522304871414
