L(s) = 1 | + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)15-s − 23-s + (0.309 − 0.951i)27-s + (−0.309 − 0.951i)31-s + (0.809 − 0.587i)37-s + (−1.61 − 1.17i)47-s + (0.309 − 0.951i)49-s + (0.618 + 1.90i)53-s + (0.809 − 0.587i)59-s − 67-s + (−0.809 − 0.587i)69-s + (−0.309 + 0.951i)71-s + (0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)15-s − 23-s + (0.309 − 0.951i)27-s + (−0.309 − 0.951i)31-s + (0.809 − 0.587i)37-s + (−1.61 − 1.17i)47-s + (0.309 − 0.951i)49-s + (0.618 + 1.90i)53-s + (0.809 − 0.587i)59-s − 67-s + (−0.809 − 0.587i)69-s + (−0.309 + 0.951i)71-s + (0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.037030106\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037030106\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−11.24233121575506221599441904429, −10.28748338170048509670783570699, −9.628471113452922889809490466949, −8.668229712712516547890205458938, −7.79809657383952038070420985908, −6.83836699581355881259483301235, −5.78023920827501190042452897403, −4.25486480901952770674551518193, −3.45010524351650637031987188390, −2.41862904401474906698208960983,
1.57343217524885922987096896120, 2.94323088331495041884507112057, 4.28718091686591138224921045754, 5.31408309055559401570823484682, 6.60722547573835642234384507499, 7.73526858035547090368709654650, 8.306740321877102695000451131511, 9.030282696148568188950768315256, 10.03175147461681629146812041804, 11.17637031849849935147176678074