| L(s) = 1 | − 0.414i·2-s − 2i·3-s + 1.82·4-s − 0.828·6-s + 4.82i·7-s − 1.58i·8-s − 9-s + 0.828·11-s − 3.65i·12-s − 2i·13-s + 1.99·14-s + 3·16-s − 2.82i·17-s + 0.414i·18-s + 4.82·19-s + ⋯ |
| L(s) = 1 | − 0.292i·2-s − 1.15i·3-s + 0.914·4-s − 0.338·6-s + 1.82i·7-s − 0.560i·8-s − 0.333·9-s + 0.249·11-s − 1.05i·12-s − 0.554i·13-s + 0.534·14-s + 0.750·16-s − 0.685i·17-s + 0.0976i·18-s + 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.71397 - 1.05929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71397 - 1.05929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.17iT - 23T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 6.48iT - 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 - 7.17iT - 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−10.28244089741735965100019592144, −9.398029793864618810230869853687, −8.352820598124405619205210920729, −7.62950753766764181203968107697, −6.64164854231641694923232928794, −6.06189807715776114358029512128, −5.06134966940389820443627324411, −3.06149903445071343546422001013, −2.42805915931343479592608498791, −1.28792416027748904210396706923,
1.45642858216634064691998303360, 3.32989835011336231261878133166, 4.03561866328231830489959466320, 4.99824140036229037522864058918, 6.21042033642033670886527893450, 7.17566378290352898151483951334, 7.67406687312108198054662755710, 8.962937722854574839775712180734, 10.10403320665227663558823477458, 10.30378047385673141809815861294
