| L(s) = 1 | + 5.65i·11-s + (−3 − 3i)13-s − 4i·19-s + (2.82 − 2.82i)23-s + 1.41·29-s + 8·31-s + (−7 + 7i)37-s + 1.41i·41-s + (4 + 4i)43-s + (2.82 + 2.82i)47-s + 7i·49-s + (8.48 − 8.48i)53-s + 11.3·59-s − 12·61-s + (−8 + 8i)67-s + ⋯ |
| L(s) = 1 | + 1.70i·11-s + (−0.832 − 0.832i)13-s − 0.917i·19-s + (0.589 − 0.589i)23-s + 0.262·29-s + 1.43·31-s + (−1.15 + 1.15i)37-s + 0.220i·41-s + (0.609 + 0.609i)43-s + (0.412 + 0.412i)47-s + i·49-s + (1.16 − 1.16i)53-s + 1.47·59-s − 1.53·61-s + (−0.977 + 0.977i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516488552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516488552\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-4 - 4i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + (8 - 8i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-3 - 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−8.636004008479413406959755916304, −7.929722585425978868505168613823, −7.06132498586174893399251663670, −6.76317249098551381625295451766, −5.56252525130084574154168737374, −4.76641505781645766909473015919, −4.34712062874239944329647649561, −2.93211582289602428292844811327, −2.37740877097733634816362643243, −1.04013909879030420156710336846,
0.51720286077981030061341818356, 1.80322266465904047654941160944, 2.90866945676150514639396811472, 3.67943489776109016588564801614, 4.56740093947293150311505605438, 5.55400025628583316592498623798, 6.04668532543660502527654888457, 7.01370058993657695691932656192, 7.61375901409437786765916076232, 8.650474593821045990736525341904
