L(s) = 1 | + (−1.77 − 1.36i)5-s + (3.33 − 3.33i)7-s + 1.66i·11-s + (−2.55 − 2.55i)13-s + (−1.21 − 1.21i)17-s − 2.64i·19-s + (−0.707 + 0.707i)23-s + (1.27 + 4.83i)25-s − 1.63·29-s + 1.29·31-s + (−10.4 + 1.35i)35-s + (5.51 − 5.51i)37-s − 10.2i·41-s + (−3.39 − 3.39i)43-s + (4.00 + 4.00i)47-s + ⋯ |
L(s) = 1 | + (−0.792 − 0.610i)5-s + (1.25 − 1.25i)7-s + 0.503i·11-s + (−0.709 − 0.709i)13-s + (−0.294 − 0.294i)17-s − 0.606i·19-s + (−0.147 + 0.147i)23-s + (0.255 + 0.966i)25-s − 0.302·29-s + 0.232·31-s + (−1.76 + 0.229i)35-s + (0.906 − 0.906i)37-s − 1.59i·41-s + (−0.518 − 0.518i)43-s + (0.583 + 0.583i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.031472485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.031472485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.77 + 1.36i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-3.33 + 3.33i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.66iT - 11T^{2} \) |
| 13 | \( 1 + (2.55 + 2.55i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.21 + 1.21i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.64iT - 19T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + (-5.51 + 5.51i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (3.39 + 3.39i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.00 - 4.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.13 - 2.13i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + (-3.61 + 3.61i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + (-3.28 - 3.28i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.68iT - 79T^{2} \) |
| 83 | \( 1 + (7.92 - 7.92i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + (7.00 - 7.00i)T - 97iT^{2} \) |
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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
−7.905491094223631354032239192640, −7.43210521603747386473240056103, −7.02118542721739031875788135303, −5.62083542304589476800204872413, −4.85551205464572122405624297870, −4.39816878399290085738387994083, −3.68218005407057761405020996612, −2.41206316283838114312834388470, −1.26560215438751436302531406200, −0.30474939579043168809457701288,
1.54576448376785254723277246101, 2.45579206272878171588062736192, 3.25370469382852128105101475715, 4.42317806142477513994928460254, 4.84063844056387383553574106169, 5.92495871547110220858705196850, 6.46176116071001962921802176749, 7.51811781816113334751091550676, 8.058124992726074145492421747809, 8.557254545483622713347935667852