L(s) = 1 | − 1.69i·3-s + (0.589 + 2.15i)5-s − 4.22i·7-s + 0.118·9-s + 4.59·11-s − 0.978i·13-s + (3.66 − 1.00i)15-s + 3.04i·17-s − 1.91·19-s − 7.17·21-s + i·23-s + (−4.30 + 2.54i)25-s − 5.29i·27-s + 0.737·29-s + 2.97·31-s + ⋯ |
L(s) = 1 | − 0.980i·3-s + (0.263 + 0.964i)5-s − 1.59i·7-s + 0.0394·9-s + 1.38·11-s − 0.271i·13-s + (0.945 − 0.258i)15-s + 0.737i·17-s − 0.439·19-s − 1.56·21-s + 0.208i·23-s + (−0.861 + 0.508i)25-s − 1.01i·27-s + 0.136·29-s + 0.535·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42678 - 1.08920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42678 - 1.08920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.589 - 2.15i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 1.69iT - 3T^{2} \) |
| 7 | \( 1 + 4.22iT - 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 0.978iT - 13T^{2} \) |
| 17 | \( 1 - 3.04iT - 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 29 | \( 1 - 0.737T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 8.93iT - 37T^{2} \) |
| 41 | \( 1 - 9.08T + 41T^{2} \) |
| 43 | \( 1 + 6.97iT - 43T^{2} \) |
| 47 | \( 1 + 2.58iT - 47T^{2} \) |
| 53 | \( 1 + 2.71iT - 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 0.731T + 61T^{2} \) |
| 67 | \( 1 - 7.16iT - 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 5.96iT - 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 - 4.39iT - 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 - 7.04iT - 97T^{2} \) |
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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
−10.14063622148230619312332195144, −9.105273641273021023785693665016, −7.84593323128568022909545978056, −7.26504814960319616038212427417, −6.64371947934869518465693872976, −6.01310022884702636271901755491, −4.24569909867365501983017787992, −3.61567465686504455629370668796, −2.07164642597222462659722304047, −0.973388223385464168789223950921,
1.51508009205966393609671318970, 2.88896906212880176879135771713, 4.27865648064786237548810794253, 4.80187812267479306740600398323, 5.80955592738332930276437125295, 6.56681537197885211116036007627, 8.064314668208737394803456869952, 9.019837344541166781318177567657, 9.246485869055617753570686231731, 9.912240985702542231350047870665