L(s) = 1 | − i·2-s + (1.30 − 1.30i)3-s − 4-s + (−1.30 − 1.30i)6-s + i·8-s − 2.41i·9-s + (−0.541 − 0.541i)11-s + (−1.30 + 1.30i)12-s + 16-s − 2.41·18-s + (−0.541 + 0.541i)22-s + (1.30 + 1.30i)24-s − i·25-s + (−1.84 − 1.84i)27-s − i·32-s − 1.41·33-s + ⋯ |
L(s) = 1 | − i·2-s + (1.30 − 1.30i)3-s − 4-s + (−1.30 − 1.30i)6-s + i·8-s − 2.41i·9-s + (−0.541 − 0.541i)11-s + (−1.30 + 1.30i)12-s + 16-s − 2.41·18-s + (−0.541 + 0.541i)22-s + (1.30 + 1.30i)24-s − i·25-s + (−1.84 − 1.84i)27-s − i·32-s − 1.41·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.530020124\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530020124\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
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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
−8.738240012030327074391859074335, −8.157901485652754734334199232637, −7.62897063622446631441577595884, −6.60507509979479617220373296874, −5.72278876903946475125999652953, −4.46631463829801752996923246455, −3.44866184685098969778769136389, −2.76874431148807665957516071096, −2.03474123137513692719430069722, −0.927467098985324896442934865830,
2.13315064029671857441814791339, 3.35807395878694034466747875955, 3.91041145608858048876225568197, 4.96093372171695306129004221416, 5.25531452446325118790529756575, 6.60827130282825585942485685934, 7.56378795872662437041138517522, 8.042765999682718482911452897786, 8.864810867888263971498066198391, 9.301537793510934717371211056326