L(s) = 1 | − 2.68i·2-s − 2.58i·3-s − 5.21·4-s + (1.90 + 1.17i)5-s − 6.93·6-s + 1.24i·7-s + 8.64i·8-s − 3.66·9-s + (3.14 − 5.11i)10-s − 5.71·11-s + 13.4i·12-s + 3.28i·13-s + 3.35·14-s + (3.02 − 4.91i)15-s + 12.7·16-s + 5.29i·17-s + ⋯ |
L(s) = 1 | − 1.89i·2-s − 1.48i·3-s − 2.60·4-s + (0.852 + 0.523i)5-s − 2.83·6-s + 0.472i·7-s + 3.05i·8-s − 1.22·9-s + (0.994 − 1.61i)10-s − 1.72·11-s + 3.88i·12-s + 0.910i·13-s + 0.897·14-s + (0.779 − 1.26i)15-s + 3.19·16-s + 1.28i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4008953523\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4008953523\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.90 - 1.17i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.68iT - 2T^{2} \) |
| 3 | \( 1 + 2.58iT - 3T^{2} \) |
| 7 | \( 1 - 1.24iT - 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 - 3.28iT - 13T^{2} \) |
| 17 | \( 1 - 5.29iT - 17T^{2} \) |
| 19 | \( 1 + 6.14T + 19T^{2} \) |
| 23 | \( 1 + 2.22iT - 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 + 4.88iT - 37T^{2} \) |
| 41 | \( 1 + 4.77T + 41T^{2} \) |
| 43 | \( 1 + 2.72iT - 43T^{2} \) |
| 47 | \( 1 + 2.52iT - 47T^{2} \) |
| 53 | \( 1 - 7.67iT - 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 - 5.27iT - 73T^{2} \) |
| 79 | \( 1 + 9.38T + 79T^{2} \) |
| 83 | \( 1 - 2.13iT - 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 10.0iT - 97T^{2} \) |
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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
−10.12766801755649994478412766443, −8.782450605764301206079333356901, −8.480028112211517217097155405615, −7.29714233079373533968463683562, −6.18406133450388293704087479598, −5.42182997699387131887506190218, −4.17023308765552351015885012932, −2.70639982405774461162075739384, −2.27549586016676459570084184629, −1.55131188772437506304282825306,
0.16658388522123664529930408105, 3.02200475190051746686696847303, 4.35114428049398482021581645347, 5.03563804479865462734752852000, 5.35731372197044688312092539890, 6.29878849031496645048183869568, 7.39835701599845912458597938994, 8.233429386047127797214750219011, 8.841810914780805853387277711397, 9.715730498785871430968861894809