L(s) = 1 | − 2-s + 4-s + (−1.72 + 2.98i)5-s − 8-s + (1.72 − 2.98i)10-s + (1 + 1.73i)11-s + (−2.44 − 4.24i)13-s + 16-s + (−1 + 1.73i)17-s + (3.72 + 6.45i)19-s + (−1.72 + 2.98i)20-s + (−1 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (−3.44 − 5.97i)25-s + (2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.771 + 1.33i)5-s − 0.353·8-s + (0.545 − 0.944i)10-s + (0.301 + 0.522i)11-s + (−0.679 − 1.17i)13-s + 0.250·16-s + (−0.242 + 0.420i)17-s + (0.854 + 1.48i)19-s + (−0.385 + 0.667i)20-s + (−0.213 − 0.369i)22-s + (−0.104 + 0.180i)23-s + (−0.689 − 1.19i)25-s + (0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2664372766\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2664372766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 6.45i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.44 + 2.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + (-1.44 + 2.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.55 - 6.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.44 - 5.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.572818265980913689762628929263, −8.230640743455473350061137556613, −7.78246275540498712698408727463, −7.28212742520703318870155562701, −6.41649716627318417922091649181, −5.70226449315397610458849061647, −4.43983029884359598304602986064, −3.38535058999238196475350437225, −2.84492031309446567306733428800, −1.57339249104882401931354646301,
0.12429021424867357496684424390, 1.12182342406409323123365894295, 2.34477638617022320976235271018, 3.58135631603707100344680741783, 4.56437875071428856707851565317, 5.10884364664703236546607619173, 6.23098562106624898520382230205, 7.24446203869520883493629907896, 7.61001255051309325360614692940, 8.669854686715264498695581561245