L(s) = 1 | + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)9-s + (0.278 + 1.76i)13-s + (1.76 + 0.896i)17-s + (0.809 − 0.587i)25-s + (−1.11 − 0.363i)29-s + (0.309 − 0.0489i)37-s + (−1.53 − 1.11i)41-s + (0.309 − 0.951i)45-s + i·49-s + (−0.809 + 0.412i)53-s + (1.53 − 1.11i)61-s + (−0.809 − 1.58i)65-s + (0.896 + 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)5-s + (−0.587 + 0.809i)9-s + (0.278 + 1.76i)13-s + (1.76 + 0.896i)17-s + (0.809 − 0.587i)25-s + (−1.11 − 0.363i)29-s + (0.309 − 0.0489i)37-s + (−1.53 − 1.11i)41-s + (0.309 − 0.951i)45-s + i·49-s + (−0.809 + 0.412i)53-s + (1.53 − 1.11i)61-s + (−0.809 − 1.58i)65-s + (0.896 + 0.142i)73-s + (−0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7696851055\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7696851055\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
good | 3 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
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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
−10.79337969074822536734365472297, −9.844811602059627396649707420494, −8.802638420795686770545355321536, −8.035109406182307324364859811192, −7.34401170687045890572021968540, −6.33394234568039182029400890812, −5.29431807238283389768293767982, −4.16026224889456025555817148412, −3.34430240467862387782471980085, −1.87574423083736883015220635774,
0.848221182522757921792540202642, 3.12553556068049031201569509690, 3.56746812148403576405076102153, 5.09476797808505959382124149576, 5.71180433670297232924542061141, 6.99419509331706234975387190193, 7.907563447797495980512772428282, 8.403673425478169050043469426016, 9.485867428759483585046104600934, 10.25934442040745597017757680956