L(s) = 1 | − 1.41i·3-s − 1.41i·5-s − 1.00·9-s − 13-s − 2.00·15-s − 17-s − 19-s + 23-s − 1.00·25-s − 29-s + 31-s + 1.41i·39-s + 41-s + 43-s + 1.41i·45-s + ⋯ |
L(s) = 1 | − 1.41i·3-s − 1.41i·5-s − 1.00·9-s − 13-s − 2.00·15-s − 17-s − 19-s + 23-s − 1.00·25-s − 29-s + 31-s + 1.41i·39-s + 41-s + 43-s + 1.41i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8987363871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8987363871\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 523 | \( 1 + T \) |
good | 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.809980212717284490792895621084, −8.114513553960013615002789782357, −7.36284662753973083263803781610, −6.69433503388423036725723554904, −5.81654499369242948325998028842, −4.90516896761321469292013783346, −4.20605710820117180850102156757, −2.57125898142421381758471277782, −1.77851909879117469308058470317, −0.61274696362115606786624164127,
2.36887899634453604565357655730, 3.02275172016426639945674680006, 4.07280201602536438451987153054, 4.64060113290736905500805193087, 5.65952728281950794161831454046, 6.61942727510073629309985975060, 7.20603144153100284069718048491, 8.220409001258977901211785666805, 9.264554314350098308956754320633, 9.654026960997043424402986950718