L(s) = 1 | + (0.809 + 0.587i)2-s − 5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)16-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)35-s + (0.309 + 0.951i)38-s + (−0.309 + 0.951i)40-s + (0.809 + 0.587i)41-s + (−0.309 + 0.951i)45-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s − 5-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (0.309 − 0.951i)14-s + (0.809 − 0.587i)16-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)35-s + (0.309 + 0.951i)38-s + (−0.309 + 0.951i)40-s + (0.809 + 0.587i)41-s + (−0.309 + 0.951i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.218300149\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218300149\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.972325190319962556273292130807, −9.545013209075178272523631630261, −8.162678412961214099815384512034, −7.36202380002155801657416394872, −6.74687668537547120398363177976, −5.90119211391729129107191851526, −4.74563035446808683017939455733, −3.91402161414175759583179288958, −3.41600072498515458727613160522, −0.990534801976033516665361488151,
2.12279685446713081119735335241, 3.07408450536182380060690743409, 3.99992463075727683850497885626, 4.92070586494791923869689276324, 5.61640950188835646776086368368, 7.03021885067196271822906544245, 7.892758070131895069074756497659, 8.500118979968597322051741501239, 9.520881951687328372734467396747, 10.62321659388754841866965603933