L(s) = 1 | − 3-s + 3·5-s − 2·9-s − 4·11-s + 5·13-s − 3·15-s − 5·17-s − 23-s + 4·25-s + 5·27-s − 3·29-s − 4·31-s + 4·33-s − 2·37-s − 5·39-s + 5·41-s − 11·43-s − 6·45-s − 5·47-s − 7·49-s + 5·51-s + 9·53-s − 12·55-s − 13·59-s − 61-s + 15·65-s + 5·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s − 1.20·11-s + 1.38·13-s − 0.774·15-s − 1.21·17-s − 0.208·23-s + 4/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.696·33-s − 0.328·37-s − 0.800·39-s + 0.780·41-s − 1.67·43-s − 0.894·45-s − 0.729·47-s − 49-s + 0.700·51-s + 1.23·53-s − 1.61·55-s − 1.69·59-s − 0.128·61-s + 1.86·65-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.608091999791254875423270667876, −7.65041563110942783967505056127, −6.51149956961343445178055326868, −6.09211697954611077691180552907, −5.43371892389953840176175891044, −4.77593602111551658490444912403, −3.46624054803871323753377357155, −2.47126883629883432592210183358, −1.59671042786130355542732378064, 0,
1.59671042786130355542732378064, 2.47126883629883432592210183358, 3.46624054803871323753377357155, 4.77593602111551658490444912403, 5.43371892389953840176175891044, 6.09211697954611077691180552907, 6.51149956961343445178055326868, 7.65041563110942783967505056127, 8.608091999791254875423270667876