L(s) = 1 | + (1.38 − 1.38i)2-s + (0.777 − 0.629i)3-s − 2.82i·4-s + (0.204 − 1.94i)6-s + (1.05 + 1.05i)7-s + (−2.52 − 2.52i)8-s + (0.207 − 0.978i)9-s + (−1.77 − 2.19i)12-s + 2.90·14-s − 4.16·16-s + (−1.29 + 1.29i)17-s + (−1.06 − 1.64i)18-s + (1.47 + 0.155i)21-s + (−3.55 − 0.373i)24-s + (−0.453 − 0.891i)27-s + (2.97 − 2.97i)28-s + ⋯ |
L(s) = 1 | + (1.38 − 1.38i)2-s + (0.777 − 0.629i)3-s − 2.82i·4-s + (0.204 − 1.94i)6-s + (1.05 + 1.05i)7-s + (−2.52 − 2.52i)8-s + (0.207 − 0.978i)9-s + (−1.77 − 2.19i)12-s + 2.90·14-s − 4.16·16-s + (−1.29 + 1.29i)17-s + (−1.06 − 1.64i)18-s + (1.47 + 0.155i)21-s + (−3.55 − 0.373i)24-s + (−0.453 − 0.891i)27-s + (2.97 − 2.97i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.531615870\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.531615870\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.777 + 0.629i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 7 | \( 1 + (-1.05 - 1.05i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1.29 - 1.29i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.831 - 0.831i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.437 + 0.437i)T + iT^{2} \) |
| 59 | \( 1 + 1.48T + T^{2} \) |
| 61 | \( 1 + 0.618T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.98iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.61iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 - 1.90T + T^{2} \) |
| 97 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
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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
−8.711892397122844264986362334213, −7.77376781651983861261487310997, −6.49080036535704880994612667654, −6.08365190710706681367860254419, −5.11663717988303952207519928377, −4.40425207178078470004345347827, −3.63667919473913796440762509899, −2.63311759109976680803684470485, −2.08091439634235423142882572127, −1.38682880217682400394396460237,
2.22940708829838400999179037081, 3.14681637824416251598279071729, 4.04951004115463760302054917732, 4.63778868908756003869332923825, 4.93260755296007592575671911599, 6.03319063803433779278225618640, 6.93991280268433854280784412942, 7.67337822935435946353536523060, 7.82802012503173054068102925567, 8.890100600078063418525390736902