L(s) = 1 | − 0.873i·3-s + (−1.89 − 1.18i)5-s − 0.992i·7-s + 2.23·9-s − 1.83·11-s + 3.28i·13-s + (−1.03 + 1.65i)15-s + 6.63i·17-s − 5.64·19-s − 0.866·21-s + i·23-s + (2.20 + 4.48i)25-s − 4.57i·27-s − 2.01·29-s + 0.315·31-s + ⋯ |
L(s) = 1 | − 0.504i·3-s + (−0.848 − 0.528i)5-s − 0.375i·7-s + 0.745·9-s − 0.552·11-s + 0.911i·13-s + (−0.266 + 0.428i)15-s + 1.60i·17-s − 1.29·19-s − 0.189·21-s + 0.208i·23-s + (0.441 + 0.897i)25-s − 0.880i·27-s − 0.374·29-s + 0.0565·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9448985695\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9448985695\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.89 + 1.18i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 0.873iT - 3T^{2} \) |
| 7 | \( 1 + 0.992iT - 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 - 3.28iT - 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 0.315T + 31T^{2} \) |
| 37 | \( 1 - 3.07iT - 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 - 5.97iT - 43T^{2} \) |
| 47 | \( 1 - 0.306iT - 47T^{2} \) |
| 53 | \( 1 + 6.98iT - 53T^{2} \) |
| 59 | \( 1 - 9.49T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 0.853iT - 67T^{2} \) |
| 71 | \( 1 + 0.797T + 71T^{2} \) |
| 73 | \( 1 - 7.67iT - 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 17.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 18.5iT - 97T^{2} \) |
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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
−9.273623967175343392723116255556, −8.293811184312895434987737977814, −7.987810192090641891659509774684, −6.96989614518871145777924780914, −6.46097467364428941624992917998, −5.25065528050156693384270988562, −4.17844846274562355976378055476, −3.87693311035970805966549097085, −2.20848990426242274659050106936, −1.20207467614542138522259820524,
0.38066344387041226806939585246, 2.32860705803556020730667019216, 3.21159634606071055738095932372, 4.16571403700587073703949059041, 4.91334980304977916071738492540, 5.82077643702022325537611346127, 7.00437837851496823354431695693, 7.43536855307595932083250885167, 8.360447473638757569873709980344, 9.090329151710506856288611403627