L(s) = 1 | − i·2-s − 4-s + (−0.707 − 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s − 1.41i·13-s − 14-s + 16-s − 1.41·19-s + (0.707 + 0.707i)20-s + 1.00i·25-s − 1.41·26-s + i·28-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.707 − 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s − 1.41i·13-s − 14-s + 16-s − 1.41·19-s + (0.707 + 0.707i)20-s + 1.00i·25-s − 1.41·26-s + i·28-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5398801684\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5398801684\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.613559001816800905084181536348, −8.035627739889890851360047391194, −7.42459472298516438936747355682, −6.16345406032983810521421051557, −5.13361758867977738076735103171, −4.43369372567518972194928568396, −3.75818322319115444571216853515, −2.90396793095614900778452243259, −1.50929048096616552089819148766, −0.36638108559426624345244067375,
2.01742807936534045086476165387, 3.22677706503195506767150351788, 4.23040721017499197367520859141, 4.80689444032175955156361529305, 6.07354163042367560984761999414, 6.41409372856590097726043341641, 7.23666946185553432540306932641, 7.981390640158427937321081169119, 8.775658579597500283025777972187, 9.179242798482861785236376953194