L(s) = 1 | + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.499 + 0.866i)12-s + (−0.499 + 0.866i)14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.499 + 0.866i)12-s + (−0.499 + 0.866i)14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.260754669\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260754669\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.681339208917920511270976386095, −9.149096458696014986153676559179, −8.405415063009880757475090522232, −7.38665100977403629311704351038, −6.81326619334976101527939302479, −5.37186191489972735021204975056, −4.12723708948269193772713983954, −3.41540471956482390613012849338, −2.37567075898719373689458010207, −1.41939505279184184132810004443,
1.73601629197791415710055944723, 2.65019234320450701681931766309, 4.05827121817273790290930100080, 5.22675251575316869647339712542, 6.08088078728970119756980616486, 6.68828012024621228645575247239, 7.79381768103158268036839742623, 8.427999182762687785448810957523, 9.235573479798853075351200299921, 9.752745901690502919007185531456