| L(s) = 1 | + (1.30 + 5.03i)3-s + (4.53 + 7.85i)5-s + (−15.6 − 9.92i)7-s + (−23.6 + 13.1i)9-s + (0.424 − 0.735i)11-s + (−21.7 + 37.6i)13-s + (−33.5 + 33.0i)15-s + (−38.3 − 66.4i)17-s + (−62.7 + 108. i)19-s + (29.5 − 91.5i)21-s + (0.0221 + 0.0383i)23-s + (21.4 − 37.0i)25-s + (−96.7 − 101. i)27-s + (−20.2 − 35.0i)29-s + 43.9·31-s + ⋯ |
| L(s) = 1 | + (0.250 + 0.968i)3-s + (0.405 + 0.702i)5-s + (−0.844 − 0.535i)7-s + (−0.874 + 0.485i)9-s + (0.0116 − 0.0201i)11-s + (−0.464 + 0.803i)13-s + (−0.578 + 0.568i)15-s + (−0.547 − 0.947i)17-s + (−0.757 + 1.31i)19-s + (0.306 − 0.951i)21-s + (0.000200 + 0.000347i)23-s + (0.171 − 0.296i)25-s + (−0.689 − 0.724i)27-s + (−0.129 − 0.224i)29-s + 0.254·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7874547698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7874547698\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 - 5.03i)T \) |
| 7 | \( 1 + (15.6 + 9.92i)T \) |
| good | 5 | \( 1 + (-4.53 - 7.85i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-0.424 + 0.735i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (21.7 - 37.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (38.3 + 66.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (62.7 - 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-0.0221 - 0.0383i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (20.2 + 35.0i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 43.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-52.7 + 91.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (208. - 360. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (169. + 293. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 580.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-235. - 408. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-411. - 713. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + 662.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-462. - 801. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (13.2 - 23.0i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-184. - 319. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.92041749048388469172008833500, −10.91013918417285638573529794661, −10.03501645384037651823914726811, −9.576619963739026916090581463615, −8.377296532879670281035066930701, −6.99813934623293116766082455790, −6.12655834150355463900746677774, −4.67318694942363463750288070460, −3.58478464382946406047422092363, −2.40991952599786366164892572540,
0.27775067390242627211066402005, 1.94473254935224216361889329155, 3.17284335023429435082226605010, 5.02778123373783136353138208886, 6.14077575953081281555055555021, 6.96620870444807851827928977482, 8.336806684501580037062020492973, 8.936590967754006634861623787525, 9.969880279433693839576652348827, 11.27491862746995586061751423060
