L(s) = 1 | + 3.60·2-s + 3.60·3-s + 8.99·4-s + 12.9·6-s + 5i·7-s + 18.0·8-s + 3.99·9-s − 10·11-s + 32.4·12-s − 3.60·13-s + 18.0i·14-s + 28.9·16-s − 15i·17-s + 14.4·18-s + (6 − 18.0i)19-s + ⋯ |
L(s) = 1 | + 1.80·2-s + 1.20·3-s + 2.24·4-s + 2.16·6-s + 0.714i·7-s + 2.25·8-s + 0.444·9-s − 0.909·11-s + 2.70·12-s − 0.277·13-s + 1.28i·14-s + 1.81·16-s − 0.882i·17-s + 0.801·18-s + (0.315 − 0.948i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(6.803261843\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.803261843\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-6 + 18.0i)T \) |
good | 2 | \( 1 - 3.60T + 4T^{2} \) |
| 3 | \( 1 - 3.60T + 9T^{2} \) |
| 7 | \( 1 - 5iT - 49T^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + 3.60T + 169T^{2} \) |
| 17 | \( 1 + 15iT - 289T^{2} \) |
| 23 | \( 1 - 35iT - 529T^{2} \) |
| 29 | \( 1 + 18.0iT - 841T^{2} \) |
| 31 | \( 1 + 36.0iT - 961T^{2} \) |
| 37 | \( 1 + 21.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 36.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 75.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 18.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 108. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 36.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 122.T + 9.40e3T^{2} \) |
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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
−11.36391024830292824329759193640, −9.926067835128235383602246565487, −8.989246667631569196899536875076, −7.80822325794979319846646500452, −7.10306209226862497409887226946, −5.72166928778086509262633407036, −5.08907327622472302920306048483, −3.84963676997206974078602232385, −2.80899820967120566905373291070, −2.30818173398764813827976122636,
1.99153945819978947106213986472, 3.04738396981920989676950008915, 3.81087305377229716940895857559, 4.80724642784834275126783989004, 5.87789743029339185414523297867, 7.00459733056120416094037082151, 7.81900525641326409971348173101, 8.750868568171052944542116320318, 10.32051056172351450305652948559, 10.73780773828025375459432670989