L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)6-s − i·8-s + 1.00i·9-s − i·11-s − 1.41·13-s − 16-s − 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (−0.707 + 0.707i)24-s + 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)6-s − i·8-s + 1.00i·9-s − i·11-s − 1.41·13-s − 16-s − 1.41i·17-s + 1.00·18-s − 22-s + i·23-s + (−0.707 + 0.707i)24-s + 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7106248893\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7106248893\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 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
−8.015369036059407892295529323352, −7.52250099827301264934006335575, −6.75595507386167244135561715019, −6.07567536730872154551572410493, −5.17447730782899358320585416819, −4.44779369337215451099611065932, −3.13744964830412146197811810835, −2.58471975704953664346923979202, −1.56079329555129479106407873404, −0.42237920486310056561746350972,
1.82915343865953018499938799369, 2.91058576601281412184481586186, 4.25940603637756066494880695018, 4.73194048346737162737136349345, 5.54764394418439074861565651129, 6.16679320257199354842584322158, 6.99197151507144015431904743570, 7.37900778656576135678451678902, 8.392519086498711585717055601627, 9.040290746226829798432368861328