L(s) = 1 | + (−0.952 + 0.303i)2-s + (0.816 − 0.577i)4-s + (−0.881 − 0.473i)5-s + (0.992 − 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.982 + 0.183i)10-s + (0.952 − 0.303i)11-s + (−0.389 + 0.920i)13-s + (−0.908 + 0.417i)14-s + (0.332 − 0.943i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.992 + 0.122i)20-s + (−0.816 + 0.577i)22-s + (−0.952 − 0.303i)23-s + ⋯ |
L(s) = 1 | + (−0.952 + 0.303i)2-s + (0.816 − 0.577i)4-s + (−0.881 − 0.473i)5-s + (0.992 − 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.982 + 0.183i)10-s + (0.952 − 0.303i)11-s + (−0.389 + 0.920i)13-s + (−0.908 + 0.417i)14-s + (0.332 − 0.943i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.992 + 0.122i)20-s + (−0.816 + 0.577i)22-s + (−0.952 − 0.303i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6901267136 - 0.7924993594i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6901267136 - 0.7924993594i\) |
\(L(1)\) |
\(\approx\) |
\(0.7102472800 - 0.09587399983i\) |
\(L(1)\) |
\(\approx\) |
\(0.7102472800 - 0.09587399983i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (-0.952 + 0.303i)T \) |
| 5 | \( 1 + (-0.881 - 0.473i)T \) |
| 7 | \( 1 + (0.992 - 0.122i)T \) |
| 11 | \( 1 + (0.952 - 0.303i)T \) |
| 13 | \( 1 + (-0.389 + 0.920i)T \) |
| 17 | \( 1 + (0.445 - 0.895i)T \) |
| 19 | \( 1 + (0.932 - 0.361i)T \) |
| 23 | \( 1 + (-0.952 - 0.303i)T \) |
| 29 | \( 1 + (-0.0307 - 0.999i)T \) |
| 31 | \( 1 + (-0.650 - 0.759i)T \) |
| 37 | \( 1 + (0.273 - 0.961i)T \) |
| 41 | \( 1 + (-0.0307 + 0.999i)T \) |
| 43 | \( 1 + (-0.969 - 0.243i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.932 + 0.361i)T \) |
| 59 | \( 1 + (0.992 + 0.122i)T \) |
| 61 | \( 1 + (0.552 + 0.833i)T \) |
| 67 | \( 1 + (-0.992 - 0.122i)T \) |
| 71 | \( 1 + (0.850 + 0.526i)T \) |
| 73 | \( 1 + (0.850 + 0.526i)T \) |
| 79 | \( 1 + (-0.0307 - 0.999i)T \) |
| 83 | \( 1 + (0.992 - 0.122i)T \) |
| 89 | \( 1 + (-0.0922 + 0.995i)T \) |
| 97 | \( 1 + (-0.998 - 0.0615i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.98391830503340174393727169486, −20.794209275389484223600880307939, −20.06305651537684152842004460029, −19.63094885526216587571016668779, −18.653737482105236104221823981531, −17.984388802892377860374620000493, −17.35629280200361934567518072684, −16.45459361352062975427460099020, −15.56523130617039739075300340908, −14.8374590803198035339187445645, −14.19527980355778263789293077019, −12.53113686877225713515414434580, −12.06220995586181963746492194036, −11.30006150221717110323283093665, −10.55586162240771092391217728337, −9.76955432717370953666617492676, −8.64081102437194511066300015563, −7.946776884776775843397150375747, −7.39919609500454274260946206698, −6.40389309026686732991347793209, −5.16994437323930046057340331739, −3.852918456367964868875034642396, −3.21274860327010849609176889743, −1.88860585174535802949179828137, −1.03803635299249104245323996970,
0.378519018094842371980523482362, 1.24578712544382854155910775841, 2.30318035274964435601110784574, 3.77158367433549353828319814653, 4.715498430142074698821747085600, 5.67196842613926346165728512013, 6.89539854887934742943066443055, 7.57679262775232254295979288164, 8.2671791262306439898582828905, 9.14547067592231469213649453816, 9.768168432454075998496617783259, 11.10655653968395052688369945280, 11.69823978586318052937612699483, 12.01357661129420289938270647724, 13.71805507465108434468346261, 14.48433721980921618983030570781, 15.15082648235660023581658458727, 16.26313698547601847475000870713, 16.561105103863499872325111316, 17.461427064044820105932898920543, 18.3244025680532738777967201345, 19.010223936471115474076997571161, 19.89704305135729949225988573689, 20.33952574010361021268838677471, 21.1936977953212685758702447606