L(s) = 1 | + (−0.415 − 0.0903i)3-s + (0.841 + 0.540i)4-s + (−1.27 + 1.10i)5-s + (−0.745 − 0.340i)9-s + (−0.654 + 0.755i)11-s + (−0.300 − 0.300i)12-s + (0.627 − 0.342i)15-s + (0.415 + 0.909i)16-s + (−1.66 + 0.239i)20-s + (−1.86 + 0.697i)23-s + (0.260 − 1.81i)25-s + (0.619 + 0.463i)27-s + (0.898 + 0.334i)31-s + (0.340 − 0.254i)33-s + (−0.442 − 0.689i)36-s + (−1.38 + 1.38i)37-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.0903i)3-s + (0.841 + 0.540i)4-s + (−1.27 + 1.10i)5-s + (−0.745 − 0.340i)9-s + (−0.654 + 0.755i)11-s + (−0.300 − 0.300i)12-s + (0.627 − 0.342i)15-s + (0.415 + 0.909i)16-s + (−1.66 + 0.239i)20-s + (−1.86 + 0.697i)23-s + (0.260 − 1.81i)25-s + (0.619 + 0.463i)27-s + (0.898 + 0.334i)31-s + (0.340 − 0.254i)33-s + (−0.442 − 0.689i)36-s + (−1.38 + 1.38i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5808481184\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5808481184\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
good | 2 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 3 | \( 1 + (0.415 + 0.0903i)T + (0.909 + 0.415i)T^{2} \) |
| 5 | \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \) |
| 7 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 13 | \( 1 + (0.909 + 0.415i)T^{2} \) |
| 17 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (0.755 - 0.654i)T^{2} \) |
| 23 | \( 1 + (1.86 - 0.697i)T + (0.755 - 0.654i)T^{2} \) |
| 29 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 31 | \( 1 + (-0.898 - 0.334i)T + (0.755 + 0.654i)T^{2} \) |
| 37 | \( 1 + (1.38 - 1.38i)T - iT^{2} \) |
| 41 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 43 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 47 | \( 1 + (-1.07 + 1.66i)T + (-0.415 - 0.909i)T^{2} \) |
| 53 | \( 1 + (-0.708 - 1.10i)T + (-0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-1.83 + 0.398i)T + (0.909 - 0.415i)T^{2} \) |
| 61 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 67 | \( 1 + (0.474 - 0.304i)T + (0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (-1.45 - 1.25i)T + (0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 97 | \( 1 + (-0.989 - 1.14i)T + (-0.142 + 0.989i)T^{2} \) |
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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
−10.58897762574744710453304450725, −10.07725241228551448304528474944, −8.417645514501517126329517884703, −7.937986259638820935720619715164, −7.04229367581401120597202539136, −6.58191222396925831811397506745, −5.46603519902056429738774384626, −4.02521267256727396398345591416, −3.25638707315663246621027119296, −2.30373759594896768913227556557,
0.53971717709319661149442619472, 2.31356689171269121995816211894, 3.64460888624384896081246865626, 4.77765835738255525872800342437, 5.56187611837839208783127126363, 6.30196754532829317000683379378, 7.60546302284848336046845140470, 8.143207688512082579457523565799, 8.876896889746122839832726162601, 10.15909836647978957558527771337