| L(s) = 1 | − 2.61i·3-s + 3.41i·5-s − 1.53·7-s − 3.82·9-s − 4.77i·11-s − 0.585i·13-s + 8.92·15-s − 2.82·17-s + 0.448i·19-s + 4i·21-s − 5.86·23-s − 6.65·25-s + 2.16i·27-s − 4.58i·29-s − 7.39·31-s + ⋯ |
| L(s) = 1 | − 1.50i·3-s + 1.52i·5-s − 0.578·7-s − 1.27·9-s − 1.44i·11-s − 0.162i·13-s + 2.30·15-s − 0.685·17-s + 0.102i·19-s + 0.872i·21-s − 1.22·23-s − 1.33·25-s + 0.416i·27-s − 0.851i·29-s − 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5557173582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5557173582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.61iT - 3T^{2} \) |
| 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 4.77iT - 11T^{2} \) |
| 13 | \( 1 + 0.585iT - 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.448iT - 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 4.58iT - 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 5.07iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2.61iT - 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 7.41iT - 53T^{2} \) |
| 59 | \( 1 - 2.61iT - 59T^{2} \) |
| 61 | \( 1 + 13.0iT - 61T^{2} \) |
| 67 | \( 1 - 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + 3.50iT - 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.538970957820693841519256121949, −8.414586430357815088355253807351, −7.67376110021305648635585992983, −6.96393556488629859217385705713, −6.21206727290853774857207538140, −5.84648956614968743383453029901, −3.78353067690999799412396844091, −2.90336035081715061912122984349, −2.01404384087698047113976413446, −0.23488572646248179933581583897,
1.85878700800801267259034683495, 3.50370917695426582723481218536, 4.47245883761770004360155368348, 4.79846791189100739025227311036, 5.76413045877705065658945560838, 6.99010377373599403184735256363, 8.202069707488906247485794507825, 8.996127297528604622609774978423, 9.603907858430836363567330753330, 9.983384594304818305806256934774
