L(s) = 1 | + (6.41 − 11.1i)5-s + (−18.4 − 31.8i)11-s − 87.1·13-s + (−51.2 − 88.8i)17-s + (47.9 − 82.9i)19-s + (−48 + 83.1i)23-s + (−19.7 − 34.1i)25-s + 212.·29-s + (−79.6 − 137. i)31-s + (−64.3 + 111. i)37-s − 298.·41-s − 33.3·43-s + (−135. + 234. i)47-s + (224. + 388. i)53-s − 472.·55-s + ⋯ |
L(s) = 1 | + (0.573 − 0.993i)5-s + (−0.504 − 0.874i)11-s − 1.85·13-s + (−0.731 − 1.26i)17-s + (0.578 − 1.00i)19-s + (−0.435 + 0.753i)23-s + (−0.157 − 0.273i)25-s + 1.35·29-s + (−0.461 − 0.799i)31-s + (−0.285 + 0.495i)37-s − 1.13·41-s − 0.118·43-s + (−0.420 + 0.728i)47-s + (0.580 + 1.00i)53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.07004986676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07004986676\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-6.41 + 11.1i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (18.4 + 31.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (51.2 + 88.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-47.9 + 82.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (48 - 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (79.6 + 137. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (64.3 - 111. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (135. - 234. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-224. - 388. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-334. - 578. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (121. - 211. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-167. - 290. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (459. + 795. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-68.1 + 118. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (80.9 - 140. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 182.T + 9.12e5T^{2} \) |
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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
−8.534762458657847439319656724681, −7.56712791661165500667682504756, −6.92211136906902551639488293991, −5.74789911969188066111380142984, −5.02650845861444621045935303205, −4.60702257484864962288527794436, −3.02536360208678009966167139979, −2.31133330988529470642714989423, −0.946298588055574099817432559619, −0.01573424633587009241594944442,
1.90729115789214363316616785034, 2.41710700516834648384205009792, 3.51746667718445164263691494161, 4.67657053184224337980990235154, 5.38487488195630563086593122026, 6.54304935545633662765765472653, 6.91690905599551012145017189151, 7.84445690669156393474188103588, 8.613203565521010074849041956638, 9.948762224367873208027991062381