| L(s) = 1 | − 1.39i·3-s + (−1.19 + 1.88i)5-s + 4.60i·7-s + 1.04·9-s − 1.56·11-s + 2.60i·13-s + (2.64 + 1.67i)15-s − 0.559i·17-s − 1.16·19-s + 6.44·21-s + i·23-s + (−2.12 − 4.52i)25-s − 5.65i·27-s + 3.17·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | − 0.807i·3-s + (−0.536 + 0.843i)5-s + 1.74i·7-s + 0.347·9-s − 0.472·11-s + 0.721i·13-s + (0.681 + 0.433i)15-s − 0.135i·17-s − 0.267·19-s + 1.40·21-s + 0.208i·23-s + (−0.424 − 0.905i)25-s − 1.08i·27-s + 0.589·29-s − 1.80·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7181762655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7181762655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.19 - 1.88i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.39iT - 3T^{2} \) |
| 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2.60iT - 13T^{2} \) |
| 17 | \( 1 + 0.559iT - 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.76iT - 43T^{2} \) |
| 47 | \( 1 - 9.32iT - 47T^{2} \) |
| 53 | \( 1 + 5.54iT - 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 - 2.60iT - 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 9.90iT - 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.71T + 89T^{2} \) |
| 97 | \( 1 - 2.62iT - 97T^{2} \) |
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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
−9.444245080725279045060008974281, −8.689280547611171568657164633632, −7.922214679205574072728820905247, −7.18156521897863206510654813042, −6.48996138755531424476649557166, −5.75533643186258221886099362815, −4.75312715731865548401248437228, −3.53367275987610326336762585147, −2.50882715273053985334232942987, −1.82760732286332101745162688280,
0.26186588378419312036299519379, 1.49331525996767358119624954116, 3.40998760491534980879845964493, 3.92454908603904624365931328644, 4.74854351960243023315157578176, 5.30721558893018936454661774595, 6.73616996183397989959744380645, 7.45956319629399072991543525317, 8.108530246152222744470515285482, 8.954478775367390916036475533844
