L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 1.70i)3-s − 1.00i·4-s + (−0.707 − 1.70i)6-s + (−0.707 − 0.707i)8-s + (−1.70 − 1.70i)9-s + (0.292 + 0.707i)11-s + (−1.70 − 0.707i)12-s − 1.00·16-s − 2.41·18-s + (0.707 + 0.292i)22-s + (−1.70 + 0.707i)24-s + (0.707 + 0.707i)25-s + (−2.41 + i)27-s + (−0.707 + 0.707i)32-s + 1.41·33-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 1.70i)3-s − 1.00i·4-s + (−0.707 − 1.70i)6-s + (−0.707 − 0.707i)8-s + (−1.70 − 1.70i)9-s + (0.292 + 0.707i)11-s + (−1.70 − 0.707i)12-s − 1.00·16-s − 2.41·18-s + (0.707 + 0.292i)22-s + (−1.70 + 0.707i)24-s + (0.707 + 0.707i)25-s + (−2.41 + i)27-s + (−0.707 + 0.707i)32-s + 1.41·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.041055147\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041055147\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (1 + i)T + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
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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
−8.973508965777866782444377268185, −7.85174700651551598378635401121, −7.21196786039913739758704243030, −6.50297816248052004954884610195, −5.82191362167887475655326859188, −4.72662067682250487955245300164, −3.61010819202117201767602411509, −2.75412850304876361118392196069, −1.96139932877948714722074248152, −1.08295442231937820697243018069,
2.57156999374206800551624852836, 3.23508923210504710431662780632, 4.11117114495189632421553558691, 4.59056626578755433530239736212, 5.53092259209884304349413457502, 6.15325166178161524119264381249, 7.35992359520098023411778699586, 8.154982115941133127642445811196, 8.918050397824901399471601729369, 9.184476988166788459305824867085