L(s) = 1 | + (0.809 − 0.587i)2-s + (0.388 + 1.19i)3-s + (0.309 − 0.951i)4-s + (−1.36 − 0.989i)5-s + (1.01 + 0.738i)6-s + (1.33 − 4.11i)7-s + (−0.309 − 0.951i)8-s + (1.14 − 0.834i)9-s − 1.68·10-s + (−1.19 − 3.09i)11-s + 1.25·12-s + (−4.20 + 3.05i)13-s + (−1.33 − 4.11i)14-s + (0.653 − 2.01i)15-s + (−0.809 − 0.587i)16-s + (5.19 + 3.77i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.224 + 0.690i)3-s + (0.154 − 0.475i)4-s + (−0.608 − 0.442i)5-s + (0.415 + 0.301i)6-s + (0.505 − 1.55i)7-s + (−0.109 − 0.336i)8-s + (0.382 − 0.278i)9-s − 0.532·10-s + (−0.361 − 0.932i)11-s + 0.362·12-s + (−1.16 + 0.847i)13-s + (−0.357 − 1.09i)14-s + (0.168 − 0.519i)15-s + (−0.202 − 0.146i)16-s + (1.26 + 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52818 - 1.07311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52818 - 1.07311i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (1.19 + 3.09i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.388 - 1.19i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (1.36 + 0.989i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-1.33 + 4.11i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (4.20 - 3.05i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.19 - 3.77i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 + (-2.19 + 6.76i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 2.03i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.48 - 4.56i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.10 - 6.48i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + (-3.72 - 11.4i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.09 - 4.43i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.21 - 12.9i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (8.31 + 6.04i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 4.78T + 67T^{2} \) |
| 71 | \( 1 + (-3.04 - 2.21i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.51 + 4.65i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.21 - 0.879i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.68 + 4.13i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 0.194T + 89T^{2} \) |
| 97 | \( 1 + (2.80 - 2.03i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.97760343628973465816004568340, −10.21421319456620402667249750192, −9.575433320422505381314317295769, −8.151278597497373671578784491472, −7.46690559330963104628805966608, −6.13195692571758538260297561312, −4.52327083774268126802372441417, −4.33480454784479948165859258331, −3.19010764907287487234156182397, −1.08662420325137212404678302115,
2.18957557511263114733952925374, 3.10257744264926081730775369335, 4.94089323110985403824876425491, 5.39373442553921352658630424588, 7.00839543746102815763093681239, 7.49892835141486714188827082498, 8.241901701391234982123200189916, 9.455588530484646167577585860151, 10.59191629768025161575802641167, 11.87118935900375393781953113163