| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 4·11-s − 6·13-s − 4·14-s − 16-s + 4·19-s − 4·22-s − 8·23-s − 6·26-s + 4·28-s − 6·29-s + 5·32-s + 6·37-s + 4·38-s + 6·43-s + 4·44-s − 8·46-s + 8·47-s + 9·49-s + 6·52-s + 12·53-s + 12·56-s − 6·58-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 1.20·11-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 0.917·19-s − 0.852·22-s − 1.66·23-s − 1.17·26-s + 0.755·28-s − 1.11·29-s + 0.883·32-s + 0.986·37-s + 0.648·38-s + 0.914·43-s + 0.603·44-s − 1.17·46-s + 1.16·47-s + 9/7·49-s + 0.832·52-s + 1.64·53-s + 1.60·56-s − 0.787·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.10885144254363, −14.50225607243793, −13.97373840386214, −13.51336959184360, −13.06784376407404, −12.54412277499523, −12.22573413611975, −11.78682607601168, −10.84529212472782, −10.19995971229563, −9.728174166968635, −9.492360364767616, −8.899991534355049, −7.910719460246979, −7.623010973213057, −6.970920552100946, −6.176414130207465, −5.672168404301718, −5.297694352344442, −4.554854584461918, −3.913399371103500, −3.368119216109212, −2.579556272709510, −2.326278889606047, −0.6304853558548080, 0,
0.6304853558548080, 2.326278889606047, 2.579556272709510, 3.368119216109212, 3.913399371103500, 4.554854584461918, 5.297694352344442, 5.672168404301718, 6.176414130207465, 6.970920552100946, 7.623010973213057, 7.910719460246979, 8.899991534355049, 9.492360364767616, 9.728174166968635, 10.19995971229563, 10.84529212472782, 11.78682607601168, 12.22573413611975, 12.54412277499523, 13.06784376407404, 13.51336959184360, 13.97373840386214, 14.50225607243793, 15.10885144254363
