L(s) = 1 | − 1.41·2-s + 1.00·4-s − 1.41i·5-s + 2.00i·10-s + (−0.707 − 0.707i)11-s − i·13-s − 0.999·16-s − 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s + 1.41i·26-s + 1.41·32-s − 1.41·41-s + (−0.707 − 0.707i)44-s + 1.41i·47-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.00·4-s − 1.41i·5-s + 2.00i·10-s + (−0.707 − 0.707i)11-s − i·13-s − 0.999·16-s − 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s + 1.41i·26-s + 1.41·32-s − 1.41·41-s + (−0.707 − 0.707i)44-s + 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3784684085\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3784684085\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 1.41T + T^{2} \) |
| 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 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
−9.349590651447221870559222340927, −8.777036335306468919375663038887, −8.055056054838760216502210306324, −7.70259549276075207132621123663, −6.36065687336432607164610453117, −5.32155161019000700325336074465, −4.62046663623492253077066128744, −3.14015175721554756128060612534, −1.67270824432691264593835114969, −0.51655883151640336112889117237,
1.81985626672467934418022898582, 2.68975853155993740255287343170, 3.99070595456108553418391363479, 5.21323718311498201761764127839, 6.73843198351514721437403768470, 6.86553503070627652387055000760, 7.80517265170894895481697674436, 8.532635996048212080025464920340, 9.548605721503100219266938815734, 10.04390623027777445219893509241