| L(s) = 1 | + 437·9-s + 242·11-s + 3.64e3·19-s − 1.60e4·29-s − 5.89e3·31-s − 1.04e3·41-s + 3.11e4·49-s + 6.35e4·59-s + 6.83e4·61-s − 2.99e4·71-s + 5.70e4·79-s + 1.31e5·81-s − 7.25e4·89-s + 1.05e5·99-s − 3.06e5·101-s + 4.66e5·109-s + 4.39e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.98e5·169-s + ⋯ |
| L(s) = 1 | + 1.79·9-s + 0.603·11-s + 2.31·19-s − 3.54·29-s − 1.10·31-s − 0.0966·41-s + 1.85·49-s + 2.37·59-s + 2.35·61-s − 0.704·71-s + 1.02·79-s + 2.23·81-s − 0.970·89-s + 1.08·99-s − 2.99·101-s + 3.76·109-s + 3/11·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.473469012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.473469012\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 437 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 31114 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 598186 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1507998 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 96 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22595 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8032 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 95 p T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 89981473 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 520 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 287836690 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 410803614 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 648207462 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 31779 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 34156 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1082368795 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14971 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2817978050 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 28538 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1874620962 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 36271 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 14694740113 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.522555311178225235240628505380, −8.971761369428194800211218926765, −8.708344793092519895417178738007, −7.82610667234566693887726709428, −7.63125761581393574153560355830, −7.15273740425347729425782677851, −7.08754443631387839601495574807, −6.61232539777846507559541582022, −5.72518767860582419641585187946, −5.43262054418980756884997467348, −5.32765824077635548425516347533, −4.48033221431959045235411847088, −3.97351512405805675860523902094, −3.62472445142636645280562348991, −3.47842994481973804572668659036, −2.36835243554766557255993276927, −2.04793364656800932560014748981, −1.35808723449671520236525505996, −1.08434319231107670095343860710, −0.42611698236531113692811863463,
0.42611698236531113692811863463, 1.08434319231107670095343860710, 1.35808723449671520236525505996, 2.04793364656800932560014748981, 2.36835243554766557255993276927, 3.47842994481973804572668659036, 3.62472445142636645280562348991, 3.97351512405805675860523902094, 4.48033221431959045235411847088, 5.32765824077635548425516347533, 5.43262054418980756884997467348, 5.72518767860582419641585187946, 6.61232539777846507559541582022, 7.08754443631387839601495574807, 7.15273740425347729425782677851, 7.63125761581393574153560355830, 7.82610667234566693887726709428, 8.708344793092519895417178738007, 8.971761369428194800211218926765, 9.522555311178225235240628505380
