L(s) = 1 | + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.891 − 0.453i)5-s + (0.453 − 0.891i)8-s + (0.587 − 0.809i)10-s + (1.76 − 0.278i)13-s + (0.809 + 0.587i)16-s + (1.04 + 0.533i)17-s + (0.707 + 0.707i)20-s + (0.587 + 0.809i)25-s + 1.78i·26-s + (−0.0966 + 0.297i)29-s + (−0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 − 1.95i)37-s + ⋯ |
L(s) = 1 | + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.891 − 0.453i)5-s + (0.453 − 0.891i)8-s + (0.587 − 0.809i)10-s + (1.76 − 0.278i)13-s + (0.809 + 0.587i)16-s + (1.04 + 0.533i)17-s + (0.707 + 0.707i)20-s + (0.587 + 0.809i)25-s + 1.78i·26-s + (−0.0966 + 0.297i)29-s + (−0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 − 1.95i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7855131875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7855131875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.891 + 0.453i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 0.533i)T + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.69 + 0.863i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (1.59 - 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
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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
−10.41877394871317287843005884868, −9.168817149496958419637572429945, −8.631393458192204959925064500083, −7.87926059231155056218986253847, −7.19576395652333267891173639853, −6.02221849244314770926248465247, −5.41170842815743080837145630179, −4.13196094150581688287690373218, −3.54955450249490258070768262134, −1.13257137602631963512333857783,
1.25331886107592805545477836199, 2.91839267484436708222363203051, 3.64984132861574759521731045714, 4.52605882809977555979158135472, 5.76627002943302944984535399193, 6.92575801427364956668583441142, 8.024484919023061344291454255322, 8.494011206983409055654279129038, 9.523028456034608609100710026638, 10.38619265595832757803448256493