| L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s + 3·9-s − 11-s + 2i·13-s − 14-s + 16-s − 6i·17-s + 3i·18-s − 4·19-s − i·22-s + 4i·23-s − 2·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 9-s − 0.301·11-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.707i·18-s − 0.917·19-s − 0.213i·22-s + 0.834i·23-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.615918528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.615918528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.479816326129258661163567921376, −8.077098416438319564800579860657, −6.93799753149081084040927248998, −6.83340271396909096252237603195, −5.81347292187717565263842944876, −4.85251568477068158071045941744, −4.50302819768858446607104191440, −3.35991804425803285052485583672, −2.32174603135251232514600516372, −1.07923888739619167603270866168,
0.54409941995117396302367316009, 1.69638798061122833127718617643, 2.54378592467189329101646827651, 3.72302765625376312485476285707, 4.21598695817485834159578869980, 5.02785724494196585416238050921, 6.04252496644771846010131421495, 6.75287921987580755101291724082, 7.65364476217957883871251322260, 8.355617627863584770411614809056
