L(s) = 1 | + (3.41 + 5.91i)5-s + (−14.9 + 10.9i)7-s + (−50.5 − 29.1i)11-s − 38.5i·13-s + (16.1 − 27.9i)17-s + (107. − 62.2i)19-s + (−174. + 100. i)23-s + (39.2 − 67.9i)25-s − 104. i·29-s + (−240. − 138. i)31-s + (−115. − 50.9i)35-s + (23.8 + 41.2i)37-s − 387.·41-s + 272.·43-s + (−81.5 − 141. i)47-s + ⋯ |
L(s) = 1 | + (0.305 + 0.528i)5-s + (−0.806 + 0.591i)7-s + (−1.38 − 0.799i)11-s − 0.822i·13-s + (0.230 − 0.398i)17-s + (1.30 − 0.751i)19-s + (−1.57 + 0.911i)23-s + (0.313 − 0.543i)25-s − 0.668i·29-s + (−1.39 − 0.805i)31-s + (−0.558 − 0.245i)35-s + (0.105 + 0.183i)37-s − 1.47·41-s + 0.966·43-s + (−0.253 − 0.438i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5147517418\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5147517418\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (14.9 - 10.9i)T \) |
good | 5 | \( 1 + (-3.41 - 5.91i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (50.5 + 29.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-16.1 + 27.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-107. + 62.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (174. - 100. i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 104. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (240. + 138. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-23.8 - 41.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (81.5 + 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (313. + 181. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (105. - 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (202. - 117. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (262. - 454. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 348. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-465. - 268. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (362. + 628. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-430. - 744. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 978. iT - 9.12e5T^{2} \) |
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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
−11.21558923701460823214868107245, −10.18748954913264819721995292070, −9.543010212336152055469179148789, −8.234259902669662799671252995632, −7.32864522455723838804126487030, −5.95271482282305792130531010673, −5.36754122893759428624318213973, −3.36962457882378480952675520333, −2.52887062492109936393277588509, −0.19151335620659905818457719205,
1.72448720054161169681059898312, 3.37789805165547307963994848489, 4.72038999887959556572151087973, 5.79933495331639190447449025713, 7.06214641273879623580167946592, 7.938505223055446101763490790933, 9.246075797432110735925416086314, 10.01179394827080594643649674832, 10.77127234192880555886782451160, 12.28868577481401689708735415592