L(s) = 1 | + (2.5 + 4.33i)7-s + (3.5 − 6.06i)13-s + 19-s + (2.5 + 4.33i)25-s + (−2 + 3.46i)31-s − 37-s + (4 + 6.92i)43-s + (−9.00 + 15.5i)49-s + (6.5 + 11.2i)61-s + (5.5 − 9.52i)67-s + 17·73-s + (−6.5 − 11.2i)79-s + 35·91-s + (−2.5 − 4.33i)97-s + (−3.5 + 6.06i)103-s + ⋯ |
L(s) = 1 | + (0.944 + 1.63i)7-s + (0.970 − 1.68i)13-s + 0.229·19-s + (0.5 + 0.866i)25-s + (−0.359 + 0.622i)31-s − 0.164·37-s + (0.609 + 1.05i)43-s + (−1.28 + 2.22i)49-s + (0.832 + 1.44i)61-s + (0.671 − 1.16i)67-s + 1.98·73-s + (−0.731 − 1.26i)79-s + 3.66·91-s + (−0.253 − 0.439i)97-s + (−0.344 + 0.597i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.927621296\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.927621296\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.5 - 4.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 17T + 73T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.611907970609641445728335825035, −8.764674666036761851865629304804, −8.280984797932974648309432722571, −7.50634895166366102485810172571, −6.18859073891285224981125564336, −5.50738124848484836225764434201, −4.93979110726106068881107107595, −3.46098609719550964888871724813, −2.57203772038842118031010330300, −1.30849831960923004611033906852,
0.967404041171581313564045931711, 2.04461999123608432021111609236, 3.80216600046040162357829510346, 4.21506353738067378167055571986, 5.19990589401770204189990802203, 6.51812222681218102349888219477, 7.04087245293201946205076287175, 7.947006811232314191575044139511, 8.666624470547837945098700561827, 9.605601286847581918190700150285