L(s) = 1 | − 6.32i·3-s + 17.8i·5-s + 22.6·7-s − 13.0·9-s + 44.2i·11-s − 17.8i·13-s + 113.·15-s + 70·17-s − 82.2i·19-s − 143. i·21-s + 158.·23-s − 195.·25-s − 88.5i·27-s + 125. i·29-s + 280·33-s + ⋯ |
L(s) = 1 | − 1.21i·3-s + 1.59i·5-s + 1.22·7-s − 0.481·9-s + 1.21i·11-s − 0.381i·13-s + 1.94·15-s + 0.998·17-s − 0.992i·19-s − 1.48i·21-s + 1.43·23-s − 1.56·25-s − 0.631i·27-s + 0.801i·29-s + 1.47·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.86733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86733\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 6.32iT - 27T^{2} \) |
| 5 | \( 1 - 17.8iT - 125T^{2} \) |
| 7 | \( 1 - 22.6T + 343T^{2} \) |
| 11 | \( 1 - 44.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 17.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 70T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 375. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 182T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 316.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 82.2iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 232. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 221. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 910T + 3.89e5T^{2} \) |
| 79 | \( 1 + 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 714. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 546T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490T + 9.12e5T^{2} \) |
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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
−12.87451666649714569753020263937, −11.80680442221122422453218854430, −10.99007053273707497041302588095, −9.949161790519663999598469526772, −8.163832432087228483075589236271, −7.22580022671621646331627904796, −6.74195622579686793005980169405, −5.00896723109477689334745653313, −2.91788851574558275541648851809, −1.58060051165689216341964464967,
1.20462889427626375237706648641, 3.79679415022851530481763655912, 4.87596700180645501866596888215, 5.57790411167408496988752528083, 7.934803100557612782905712947515, 8.730562439661298931194605571401, 9.582775670188518468128407765530, 10.82447452772290597494812629115, 11.70129097841085557184618340537, 12.81146615570764980042204002957