L(s) = 1 | − 2.42i·2-s − 1.05i·3-s − 3.89·4-s + (0.123 + 2.23i)5-s − 2.56·6-s − 4.52i·7-s + 4.59i·8-s + 1.88·9-s + (5.41 − 0.299i)10-s + 0.194·11-s + 4.11i·12-s + 3.48i·13-s − 10.9·14-s + (2.35 − 0.130i)15-s + 3.36·16-s − 6.60i·17-s + ⋯ |
L(s) = 1 | − 1.71i·2-s − 0.610i·3-s − 1.94·4-s + (0.0551 + 0.998i)5-s − 1.04·6-s − 1.71i·7-s + 1.62i·8-s + 0.627·9-s + (1.71 − 0.0946i)10-s + 0.0586·11-s + 1.18i·12-s + 0.967i·13-s − 2.93·14-s + (0.609 − 0.0336i)15-s + 0.840·16-s − 1.60i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.009361904\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009361904\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.123 - 2.23i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.42iT - 2T^{2} \) |
| 3 | \( 1 + 1.05iT - 3T^{2} \) |
| 7 | \( 1 + 4.52iT - 7T^{2} \) |
| 11 | \( 1 - 0.194T + 11T^{2} \) |
| 13 | \( 1 - 3.48iT - 13T^{2} \) |
| 17 | \( 1 + 6.60iT - 17T^{2} \) |
| 23 | \( 1 + 1.18iT - 23T^{2} \) |
| 29 | \( 1 + 4.10T + 29T^{2} \) |
| 31 | \( 1 + 5.06T + 31T^{2} \) |
| 37 | \( 1 + 4.20iT - 37T^{2} \) |
| 41 | \( 1 + 9.20T + 41T^{2} \) |
| 43 | \( 1 + 2.26iT - 43T^{2} \) |
| 47 | \( 1 + 5.95iT - 47T^{2} \) |
| 53 | \( 1 - 6.94iT - 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 + 5.22iT - 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 + 6.54iT - 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 - 9.54iT - 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 7.41iT - 97T^{2} \) |
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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
−9.172152369865921250768539332130, −7.74471226114101440959801040383, −7.09162203973035798510947976704, −6.67710701324565617354988495165, −4.99591971708064643746111507678, −4.03053664231171720689987245727, −3.56159129239665729455184371246, −2.37821466315359800371746047657, −1.53768971657821667013214081041, −0.38035134702239371389049874510,
1.77046822498661603395191794661, 3.53851442172795864971454798987, 4.52382786196445243695365723694, 5.36053618292661975721715378849, 5.65083962188155322403972502418, 6.48649134714761267855778537459, 7.64015589288090585805440143331, 8.383341576194444212276197400167, 8.744173892313296055282188224795, 9.495848699797589934042903810870