| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 2·26-s + 2·29-s − 8·31-s − 32-s − 2·34-s + 37-s + 4·38-s − 40-s + 10·41-s + 12·43-s − 4·44-s − 8·46-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.164·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.26413609283194, −12.99609337873068, −12.65475127272625, −12.15956370138957, −11.27063415758647, −11.00238038576766, −10.67401189482640, −10.26446934518221, −9.512369378681659, −9.286306057854770, −8.685047071515336, −8.306216487171595, −7.647818156047886, −7.327778461316447, −6.766825882618436, −6.142565260607807, −5.631185657416643, −5.300426593209351, −4.556334364976770, −3.951364180315047, −3.206158747272081, −2.616869645477461, −2.275789694289695, −1.385085079589816, −0.8612578122161018, 0,
0.8612578122161018, 1.385085079589816, 2.275789694289695, 2.616869645477461, 3.206158747272081, 3.951364180315047, 4.556334364976770, 5.300426593209351, 5.631185657416643, 6.142565260607807, 6.766825882618436, 7.327778461316447, 7.647818156047886, 8.306216487171595, 8.685047071515336, 9.286306057854770, 9.512369378681659, 10.26446934518221, 10.67401189482640, 11.00238038576766, 11.27063415758647, 12.15956370138957, 12.65475127272625, 12.99609337873068, 13.26413609283194
