L(s) = 1 | + (0.587 + 0.809i)5-s + (0.951 − 0.309i)9-s + (−0.142 + 0.278i)13-s + (0.896 + 0.142i)17-s + (−0.309 + 0.951i)25-s + (−0.5 − 0.363i)29-s + (0.809 + 0.412i)37-s + (−0.363 − 1.11i)41-s + (0.809 + 0.587i)45-s − i·49-s + (0.309 + 1.95i)53-s + (−1.53 − 0.5i)61-s + (−0.309 + 0.0489i)65-s + (−0.142 − 0.278i)73-s + (0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)5-s + (0.951 − 0.309i)9-s + (−0.142 + 0.278i)13-s + (0.896 + 0.142i)17-s + (−0.309 + 0.951i)25-s + (−0.5 − 0.363i)29-s + (0.809 + 0.412i)37-s + (−0.363 − 1.11i)41-s + (0.809 + 0.587i)45-s − i·49-s + (0.309 + 1.95i)53-s + (−1.53 − 0.5i)61-s + (−0.309 + 0.0489i)65-s + (−0.142 − 0.278i)73-s + (0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.535554039\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535554039\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
good | 3 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
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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
−9.236765117056469121325257265614, −7.967550317849732960162026461080, −7.38653901625875500025771548640, −6.66599129740851943085304256155, −6.00926090773528396167299108451, −5.17696595759226128560425953287, −4.13953179564963238509685793674, −3.37625404651925308108995833512, −2.35582403769640637779959003439, −1.37135749251995518460145877372,
1.11636081852424014317510412329, 2.01208108204101234619926427325, 3.19064604361024945934384682474, 4.27647695263755500911246797306, 4.94592735886950732673374148132, 5.66775906828682165187345455829, 6.45902644477647715307552933550, 7.44520579311409667292222986724, 7.965089829694433965207288900676, 8.842689717649587886316908821118