L(s) = 1 | + (−0.955 + 0.294i)4-s + (−0.365 + 0.930i)7-s + (0.167 + 1.11i)13-s + (0.826 − 0.563i)16-s + (1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.0747 − 0.997i)28-s + (−1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (0.147 + 1.97i)43-s + (−0.733 − 0.680i)49-s + (−0.488 − 1.01i)52-s + (−0.173 + 0.563i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 − 0.129i)67-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)4-s + (−0.365 + 0.930i)7-s + (0.167 + 1.11i)13-s + (0.826 − 0.563i)16-s + (1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.0747 − 0.997i)28-s + (−1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (0.147 + 1.97i)43-s + (−0.733 − 0.680i)49-s + (−0.488 − 1.01i)52-s + (−0.173 + 0.563i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 − 0.129i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7843007217\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7843007217\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.365 - 0.930i)T \) |
good | 2 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (-0.167 - 1.11i)T + (-0.955 + 0.294i)T^{2} \) |
| 17 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 19 | \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 31 | \( 1 + (1.17 - 0.680i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.07 - 0.997i)T + (0.0747 - 0.997i)T^{2} \) |
| 41 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 43 | \( 1 + (-0.147 - 1.97i)T + (-0.988 + 0.149i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.563i)T + (-0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (1.81 - 0.712i)T + (0.733 - 0.680i)T^{2} \) |
| 79 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 89 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 97 | \( 1 + (0.258 - 0.149i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.821440919486837036574882549647, −8.423544655168286723570486495339, −7.38255420733399915711455207613, −6.70988235977411229986595753534, −5.79414959744181509442700759133, −5.04184904230205882585813330333, −4.43589431695753973734189666490, −3.38961171421906460267116243551, −2.73415781699611075637741205391, −1.36411242775700653958271615671,
0.49910378846716486912532331841, 1.61488707150331169677926293600, 3.33356276553011146153534198468, 3.61610315987856279854662114898, 4.65240501456672409390431872915, 5.51266327822160087136090421902, 5.89063884801047376011873370998, 7.30348114594009785247241372793, 7.46282794213113685774996498449, 8.517057102574292304947683540542