| L(s) = 1 | − 2-s + 3-s − 4-s + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s − 12-s − 2·13-s + 2·15-s − 16-s − 2·17-s − 18-s − 2·20-s + 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s + 6·29-s − 2·30-s + 8·31-s − 5·32-s + 2·34-s − 36-s + 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s − 0.365·30-s + 1.43·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044012802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044012802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.97485726452203, −15.07498365087478, −14.78642022042977, −14.07973935457637, −13.57461274353071, −13.22885172798431, −12.76106503611736, −11.89356906246398, −11.27012166972771, −10.37428056260053, −10.16201132830474, −9.532907469406108, −9.103289507782321, −8.531873237522727, −8.043417754758536, −7.311030202032632, −6.716525565339301, −6.049327383188480, −5.025371784402288, −4.773410203600274, −3.937325413026200, −2.935579789055321, −2.369737489874612, −1.456967865338094, −0.7045650789747930,
0.7045650789747930, 1.456967865338094, 2.369737489874612, 2.935579789055321, 3.937325413026200, 4.773410203600274, 5.025371784402288, 6.049327383188480, 6.716525565339301, 7.311030202032632, 8.043417754758536, 8.531873237522727, 9.103289507782321, 9.532907469406108, 10.16201132830474, 10.37428056260053, 11.27012166972771, 11.89356906246398, 12.76106503611736, 13.22885172798431, 13.57461274353071, 14.07973935457637, 14.78642022042977, 15.07498365087478, 15.97485726452203
