| L(s) = 1 | + 3i·3-s − 8i·7-s − 9·9-s − 20·11-s − 22i·13-s − 14i·17-s + 76·19-s + 24·21-s + 56i·23-s − 27i·27-s + 154·29-s − 160·31-s − 60i·33-s − 162i·37-s + 66·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.431i·7-s − 0.333·9-s − 0.548·11-s − 0.469i·13-s − 0.199i·17-s + 0.917·19-s + 0.249·21-s + 0.507i·23-s − 0.192i·27-s + 0.986·29-s − 0.926·31-s − 0.316i·33-s − 0.719i·37-s + 0.270·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7049510378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7049510378\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8iT - 343T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 76T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 154T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 544iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 210iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 380T + 2.05e5T^{2} \) |
| 61 | \( 1 + 794T + 2.26e5T^{2} \) |
| 67 | \( 1 - 148iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 840T + 3.57e5T^{2} \) |
| 73 | \( 1 + 858iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 144T + 4.93e5T^{2} \) |
| 83 | \( 1 - 316iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994iT - 9.12e5T^{2} \) |
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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
−9.703761449503972726619320759788, −9.087996551939526135271441453945, −7.975258243894503479579693685584, −7.45050294778172252186820975093, −6.30215174995228536767955346536, −5.36582272938735196608363762073, −4.64670266121736860728973935081, −3.54908475668306638398458777956, −2.73979985463401502677564988241, −1.19746091700552103153788104231,
0.17250046632785478178348295049, 1.55861527328299998527599316241, 2.56583617449516758689176651474, 3.59668455862464165243641466471, 4.91050235297953364896710394447, 5.62821136675522683767107550213, 6.64410314730406484645451238306, 7.29416704266608571218871708944, 8.292214279890669564797985451472, 8.830491345389923760241372511374
