L(s) = 1 | − 2-s + 3.42·3-s + 4-s + 5-s − 3.42·6-s + 2.68·7-s − 8-s + 8.74·9-s − 10-s − 11-s + 3.42·12-s − 5.61·13-s − 2.68·14-s + 3.42·15-s + 16-s + 6.29·17-s − 8.74·18-s + 2.23·19-s + 20-s + 9.19·21-s + 22-s − 7.18·23-s − 3.42·24-s + 25-s + 5.61·26-s + 19.6·27-s + 2.68·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s + 1.01·7-s − 0.353·8-s + 2.91·9-s − 0.316·10-s − 0.301·11-s + 0.989·12-s − 1.55·13-s − 0.717·14-s + 0.884·15-s + 0.250·16-s + 1.52·17-s − 2.06·18-s + 0.512·19-s + 0.223·20-s + 2.00·21-s + 0.213·22-s − 1.49·23-s − 0.699·24-s + 0.200·25-s + 1.10·26-s + 3.78·27-s + 0.507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.251235780\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.251235780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 - 5.79T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 79 | \( 1 + 0.450T + 79T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.023528140309215873746517130635, −7.39898545137320695792084517438, −7.03870739966632181703741958394, −5.72092079525853225842063221113, −4.92858906705733558842197781902, −4.11823043502504668591528906016, −3.19572765636289200410362080885, −2.47707680147178938963551558976, −1.95102959829321255430758189257, −1.10475980067357862224543207354,
1.10475980067357862224543207354, 1.95102959829321255430758189257, 2.47707680147178938963551558976, 3.19572765636289200410362080885, 4.11823043502504668591528906016, 4.92858906705733558842197781902, 5.72092079525853225842063221113, 7.03870739966632181703741958394, 7.39898545137320695792084517438, 8.023528140309215873746517130635