L(s) = 1 | − i·3-s + 4i·7-s − 9-s + 6i·13-s − 2i·17-s − 4·19-s + 4·21-s + 8i·23-s + i·27-s + 6·29-s − 6i·37-s + 6·39-s + 10·41-s + 4i·43-s + 8i·47-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.66i·13-s − 0.485i·17-s − 0.917·19-s + 0.872·21-s + 1.66i·23-s + 0.192i·27-s + 1.11·29-s − 0.986i·37-s + 0.960·39-s + 1.56·41-s + 0.609i·43-s + 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06391 + 0.657533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06391 + 0.657533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−11.18524216341932078040356757274, −9.638606927349785461511499718977, −9.078807256848167901194755803226, −8.287448057278562022482531391962, −7.18100296939035834821071657271, −6.29720101906815292738912775970, −5.49193846641471260402383809720, −4.28788880917770057848516441547, −2.73977795103903345474604748271, −1.78067278372300495508591919348,
0.70899130635208314787033985535, 2.79134621649960698037904359785, 3.96788115327728612406976224596, 4.70067279448851147077453764998, 5.95738066798228525721165691884, 6.93180134066851249310875348524, 7.996633054832418871659441165344, 8.623172100411390454070304948476, 10.03990665123640350444670335975, 10.46690162467379233142421946017