| L(s) = 1 | − 2·2-s + 4-s − 2·5-s − 2·7-s + 4·10-s + 2·11-s + 2·13-s + 4·14-s + 16-s − 4·17-s + 6·19-s − 2·20-s − 4·22-s − 4·23-s + 3·25-s − 4·26-s − 2·28-s + 8·29-s + 12·31-s + 2·32-s + 8·34-s + 4·35-s − 12·38-s + 10·41-s − 2·43-s + 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 1.26·10-s + 0.603·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.447·20-s − 0.852·22-s − 0.834·23-s + 3/5·25-s − 0.784·26-s − 0.377·28-s + 1.48·29-s + 2.15·31-s + 0.353·32-s + 1.37·34-s + 0.676·35-s − 1.94·38-s + 1.56·41-s − 0.304·43-s + 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6846547483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6846547483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 151 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 121 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 95 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−10.02792358613329028896778447612, −9.756276417316992312531199280960, −9.399618859337813692756970664894, −9.027239797356005786129952452974, −8.572370578576987369747785803198, −8.349047094470360653597024532452, −7.75991674089887446705496511930, −7.67075274733825648820178862853, −6.86695097279856306167237730752, −6.49157910502769604567010324267, −6.26244710271478449709489221025, −5.67617466705980641713739976336, −4.75650797871978601491027775130, −4.57129197376866602325310791351, −3.88628529066027400245505538327, −3.39046012594370553455550625182, −2.86280651394709953847482887950, −2.16753262769038523831216053910, −0.878116091487073529779634856287, −0.73557699812010142157301659130,
0.73557699812010142157301659130, 0.878116091487073529779634856287, 2.16753262769038523831216053910, 2.86280651394709953847482887950, 3.39046012594370553455550625182, 3.88628529066027400245505538327, 4.57129197376866602325310791351, 4.75650797871978601491027775130, 5.67617466705980641713739976336, 6.26244710271478449709489221025, 6.49157910502769604567010324267, 6.86695097279856306167237730752, 7.67075274733825648820178862853, 7.75991674089887446705496511930, 8.349047094470360653597024532452, 8.572370578576987369747785803198, 9.027239797356005786129952452974, 9.399618859337813692756970664894, 9.756276417316992312531199280960, 10.02792358613329028896778447612
