| L(s) = 1 | − 2.07e18·2-s − 2.64e28·3-s + 1.63e36·4-s − 5.16e41·5-s + 5.48e46·6-s − 2.39e51·7-s + 2.11e54·8-s − 4.69e57·9-s + 1.06e60·10-s + 6.81e62·11-s − 4.32e64·12-s + 2.56e66·13-s + 4.97e69·14-s + 1.36e70·15-s − 8.73e72·16-s − 1.61e74·17-s + 9.72e75·18-s − 1.10e76·19-s − 8.44e77·20-s + 6.34e79·21-s − 1.41e81·22-s − 5.71e81·23-s − 5.60e82·24-s − 3.49e84·25-s − 5.32e84·26-s + 2.66e86·27-s − 3.92e87·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.360·3-s + 0.615·4-s − 0.266·5-s + 0.458·6-s − 1.78·7-s + 0.488·8-s − 0.870·9-s + 0.338·10-s + 0.674·11-s − 0.221·12-s + 0.103·13-s + 2.26·14-s + 0.0959·15-s − 1.23·16-s − 0.584·17-s + 1.10·18-s − 0.0476·19-s − 0.163·20-s + 0.643·21-s − 0.857·22-s − 0.235·23-s − 0.176·24-s − 0.929·25-s − 0.131·26-s + 0.673·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(122-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+60.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(61)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{123}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.07e18T + 2.65e36T^{2} \) |
| 3 | \( 1 + 2.64e28T + 5.39e57T^{2} \) |
| 5 | \( 1 + 5.16e41T + 3.76e84T^{2} \) |
| 7 | \( 1 + 2.39e51T + 1.80e102T^{2} \) |
| 11 | \( 1 - 6.81e62T + 1.01e126T^{2} \) |
| 13 | \( 1 - 2.56e66T + 6.12e134T^{2} \) |
| 17 | \( 1 + 1.61e74T + 7.66e148T^{2} \) |
| 19 | \( 1 + 1.10e76T + 5.36e154T^{2} \) |
| 23 | \( 1 + 5.71e81T + 5.87e164T^{2} \) |
| 29 | \( 1 - 5.46e88T + 8.91e176T^{2} \) |
| 31 | \( 1 + 1.61e90T + 2.84e180T^{2} \) |
| 37 | \( 1 + 2.70e94T + 5.65e189T^{2} \) |
| 41 | \( 1 - 3.14e97T + 1.40e195T^{2} \) |
| 43 | \( 1 - 7.06e98T + 4.46e197T^{2} \) |
| 47 | \( 1 - 1.46e101T + 2.10e202T^{2} \) |
| 53 | \( 1 - 3.40e104T + 4.33e208T^{2} \) |
| 59 | \( 1 - 1.01e107T + 1.87e214T^{2} \) |
| 61 | \( 1 + 6.56e107T + 1.05e216T^{2} \) |
| 67 | \( 1 + 3.87e110T + 9.01e220T^{2} \) |
| 71 | \( 1 - 4.17e110T + 1.00e224T^{2} \) |
| 73 | \( 1 - 8.94e112T + 2.89e225T^{2} \) |
| 79 | \( 1 + 8.25e114T + 4.10e229T^{2} \) |
| 83 | \( 1 - 2.01e116T + 1.61e232T^{2} \) |
| 89 | \( 1 + 1.38e118T + 7.51e235T^{2} \) |
| 97 | \( 1 + 2.29e120T + 2.50e240T^{2} \) |
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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
−12.15274890577328502682421208731, −10.71492008105037698929704646682, −9.531015550964795101385362792045, −8.635635453191939513376134751572, −7.04511979602025020198100574335, −5.99513591697079199126525767723, −3.98698517495385285832923279003, −2.55081171416203299290435805325, −0.796403657589990784788379447132, 0,
0.796403657589990784788379447132, 2.55081171416203299290435805325, 3.98698517495385285832923279003, 5.99513591697079199126525767723, 7.04511979602025020198100574335, 8.635635453191939513376134751572, 9.531015550964795101385362792045, 10.71492008105037698929704646682, 12.15274890577328502682421208731
