L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 16-s − 2·22-s + 25-s − 32-s − 2·43-s + 2·44-s − 50-s + 64-s + 2·67-s + 2·86-s − 2·88-s + 100-s − 2·107-s + 2·113-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s + 16-s − 2·22-s + 25-s − 32-s − 2·43-s + 2·44-s − 50-s + 64-s + 2·67-s + 2·86-s − 2·88-s + 100-s − 2·107-s + 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9479227530\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9479227530\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.706261240350169393988595552389, −8.285981089508887176764867615008, −7.17570084857949925488445180970, −6.70778009095661027703257213759, −6.11102713147333405931398758047, −5.03051945112957078558261094584, −3.91623062565850856913642172481, −3.15156071375818914698984792631, −1.93212856658993978577026009985, −1.05447110805962188879110148449,
1.05447110805962188879110148449, 1.93212856658993978577026009985, 3.15156071375818914698984792631, 3.91623062565850856913642172481, 5.03051945112957078558261094584, 6.11102713147333405931398758047, 6.70778009095661027703257213759, 7.17570084857949925488445180970, 8.285981089508887176764867615008, 8.706261240350169393988595552389