L(s) = 1 | + 3.25i·3-s + (1.49 − 1.66i)5-s − i·7-s − 7.57·9-s + 2.88·11-s − 6.94i·13-s + (5.41 + 4.86i)15-s − 0.549i·17-s − 4.98·19-s + 3.25·21-s + 2.33i·23-s + (−0.533 − 4.97i)25-s − 14.8i·27-s − 5.14·29-s − 5.40·31-s + ⋯ |
L(s) = 1 | + 1.87i·3-s + (0.668 − 0.743i)5-s − 0.377i·7-s − 2.52·9-s + 0.870·11-s − 1.92i·13-s + (1.39 + 1.25i)15-s − 0.133i·17-s − 1.14·19-s + 0.709·21-s + 0.487i·23-s + (−0.106 − 0.994i)25-s − 2.86i·27-s − 0.954·29-s − 0.970·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.313371035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313371035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.49 + 1.66i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 3.25iT - 3T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + 6.94iT - 13T^{2} \) |
| 17 | \( 1 + 0.549iT - 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 2.33iT - 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 8.31iT - 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 - 1.81iT - 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 - 2.18iT - 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 3.38iT - 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 + 5.22iT - 97T^{2} \) |
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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
−9.000325175949536512759765906336, −8.620788093155006874222961787724, −7.60864445731455838934300155360, −6.23248099358959799930294678933, −5.43759688025765945762596603652, −5.09011015316082014941943007053, −3.92940730040087000405542344986, −3.57854830357646789195008587773, −2.21566519932586755694601591781, −0.42562627953575604113097304527,
1.55535859624896390200128746649, 1.96970008729663027468946642195, 2.92187667982185873330877949388, 4.20872055386010354021942921472, 5.60457596941308117476262977549, 6.43928597414847067484379786339, 6.63064141086741522611670340386, 7.29588317165944197100813628530, 8.329658422412572274369603818588, 8.975364897478225551006350079226