L(s) = 1 | + (−2.57 − 2.57i)5-s + 3.74·7-s + (3.64 − 3.64i)11-s + (−4.64 − 4.64i)13-s − 0.913i·17-s + (−1.41 + 1.41i)19-s − 6i·23-s + 8.29i·25-s + (1.66 − 1.66i)29-s + 6.57i·31-s + (−9.64 − 9.64i)35-s + (−1 + i)37-s + 0.913·41-s + (3.74 + 3.74i)43-s − 6·47-s + ⋯ |
L(s) = 1 | + (−1.15 − 1.15i)5-s + 1.41·7-s + (1.09 − 1.09i)11-s + (−1.28 − 1.28i)13-s − 0.221i·17-s + (−0.324 + 0.324i)19-s − 1.25i·23-s + 1.65i·25-s + (0.309 − 0.309i)29-s + 1.18i·31-s + (−1.63 − 1.63i)35-s + (−0.164 + 0.164i)37-s + 0.142·41-s + (0.570 + 0.570i)43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.250515352\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250515352\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.57 + 2.57i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.64 + 4.64i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.913iT - 17T^{2} \) |
| 19 | \( 1 + (1.41 - 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (-1.66 + 1.66i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.57iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.913T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (-3.49 - 3.49i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.29 - 1.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.291 - 0.291i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.32 - 3.32i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 8.58iT - 73T^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (-4.93 - 4.93i)T + 83iT^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.551976545757529240678102573999, −8.024146520067301027032769031637, −7.50408049566899076251476460688, −6.31334138843863904712726104245, −5.19693736368097726430557775423, −4.74520817741063687280873920474, −3.99307778760598856693753137001, −2.89412801163881364953531421641, −1.38117923257569081981834958958, −0.45523259228821677282474347905,
1.65913639262212216592095897357, 2.48750815081801553129709224391, 3.92937506336165295895033447480, 4.28562623039739060467603419157, 5.16420763558289917539038473922, 6.55174996268284690062109335636, 7.19826873298542591504155823373, 7.52727159009243262716996271864, 8.411759484151078865506276478396, 9.371041643527694385300480711170