L(s) = 1 | + (−0.363 + 1.11i)2-s + (1.53 − 1.11i)3-s + (−0.309 − 0.224i)4-s + (0.309 + 0.951i)5-s + (0.690 + 2.12i)6-s + (−0.587 + 0.427i)8-s + (0.809 − 2.48i)9-s − 1.17·10-s + (−0.309 + 0.951i)11-s − 0.726·12-s + (0.363 − 1.11i)13-s + (1.53 + 1.11i)15-s + (−0.381 − 1.17i)16-s + (2.48 + 1.80i)18-s + (−0.809 + 0.587i)19-s + (0.118 − 0.363i)20-s + ⋯ |
L(s) = 1 | + (−0.363 + 1.11i)2-s + (1.53 − 1.11i)3-s + (−0.309 − 0.224i)4-s + (0.309 + 0.951i)5-s + (0.690 + 2.12i)6-s + (−0.587 + 0.427i)8-s + (0.809 − 2.48i)9-s − 1.17·10-s + (−0.309 + 0.951i)11-s − 0.726·12-s + (0.363 − 1.11i)13-s + (1.53 + 1.11i)15-s + (−0.381 − 1.17i)16-s + (2.48 + 1.80i)18-s + (−0.809 + 0.587i)19-s + (0.118 − 0.363i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.449433404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449433404\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 1.90T + T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−9.883198261832472419808319146747, −9.083347724061209531561929943101, −8.220256903732915233898535894568, −7.75261640013853169262928348867, −7.06494637740494984434123344186, −6.51533449898577233875184938146, −5.58986019817021147651563329516, −3.71374492694138558055417172810, −2.77123326125437657688199875683, −1.97159446593751785041650031403,
1.67847950852051510447841201220, 2.56314855092256693030614263218, 3.57075966303427258618753091771, 4.28919572304668309142122561397, 5.30790293187428848139580646762, 6.69709130692282039604596655501, 8.263287303387186048776398538977, 8.665557037628344088807759278672, 9.213191786333649739786144049378, 9.883709301151011800272735872998