L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.190 + 0.587i)13-s + (0.309 − 0.951i)16-s + (−1.30 − 0.951i)17-s − 0.999·18-s + 0.618·26-s + (1.30 − 0.951i)29-s − 32-s + (−0.499 + 1.53i)34-s + (0.309 + 0.951i)36-s + (0.5 − 1.53i)37-s + (0.190 − 0.587i)41-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.190 + 0.587i)13-s + (0.309 − 0.951i)16-s + (−1.30 − 0.951i)17-s − 0.999·18-s + 0.618·26-s + (1.30 − 0.951i)29-s − 32-s + (−0.499 + 1.53i)34-s + (0.309 + 0.951i)36-s + (0.5 − 1.53i)37-s + (0.190 − 0.587i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8415761528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8415761528\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.100791510471794249471117039380, −8.376465548855396364100881401055, −7.31553408366207425601812969308, −6.73445601922226179915284642560, −5.63238755456889284065269091451, −4.45178045845089100633717738064, −4.08078958206713210802183495819, −2.88275204725785428114534985480, −2.08711285937985527778426775329, −0.67495917914578661225401303381,
1.38158039865059343066219962935, 2.66874459260068934211052937482, 4.09836404837452125144220482350, 4.76990885636083203532849094261, 5.51134884825536185985176763639, 6.46895173107416869387732625773, 6.99778756533648438635666088023, 8.005772522897979600576685363862, 8.345008201121621420508644503123, 9.158547463739272614864621414132