L(s) = 1 | − 0.317i·5-s + i·7-s + 1.41·11-s − 4.44·13-s − 5.97i·17-s + 2.44i·19-s − 5.65·23-s + 4.89·25-s + 5.51i·29-s − 6.44i·31-s + 0.317·35-s + 9·37-s + 7.24i·41-s + 6.34i·43-s − 10.8·47-s + ⋯ |
L(s) = 1 | − 0.142i·5-s + 0.377i·7-s + 0.426·11-s − 1.23·13-s − 1.44i·17-s + 0.561i·19-s − 1.17·23-s + 0.979·25-s + 1.02i·29-s − 1.15i·31-s + 0.0537·35-s + 1.47·37-s + 1.13i·41-s + 0.968i·43-s − 1.58·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6274522668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6274522668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.317iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.97iT - 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.51iT - 29T^{2} \) |
| 31 | \( 1 + 6.44iT - 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 - 7.24iT - 41T^{2} \) |
| 43 | \( 1 - 6.34iT - 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 14.3iT - 79T^{2} \) |
| 83 | \( 1 - 8.02T + 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.134129382718348448390291473053, −7.79168154611401389226191866399, −6.84567328571023144385413617798, −6.33107897416134227291517370974, −5.32553435020339519115285291831, −4.84470315280512606941124817371, −4.02372797656701503200510039004, −2.95279987940488461058959726773, −2.34035872810408318314180758878, −1.16258443914382381030516145491,
0.16263994893559228941018338420, 1.51267564013525909546234150124, 2.41809082446386980084621520154, 3.34584182821393329618124526860, 4.24263099620280640355717897319, 4.74937857605992617712100664139, 5.79698560399343750598438187507, 6.39171373799039633449429844258, 7.14536998319702744695221369150, 7.74800446019954675979808087438