L(s) = 1 | − 3.30·3-s − 5-s − 0.302·7-s + 7.90·9-s + 5.30·11-s + 0.302·13-s + 3.30·15-s − 3.90·17-s + 4.90·19-s + 1.00·21-s − 23-s + 25-s − 16.2·27-s − 4.60·29-s + 2.90·31-s − 17.5·33-s + 0.302·35-s − 8·37-s − 1.00·39-s − 9.90·41-s − 5.21·43-s − 7.90·45-s + 4.60·47-s − 6.90·49-s + 12.9·51-s − 3.21·53-s − 5.30·55-s + ⋯ |
L(s) = 1 | − 1.90·3-s − 0.447·5-s − 0.114·7-s + 2.63·9-s + 1.59·11-s + 0.0839·13-s + 0.852·15-s − 0.947·17-s + 1.12·19-s + 0.218·21-s − 0.208·23-s + 0.200·25-s − 3.11·27-s − 0.855·29-s + 0.522·31-s − 3.04·33-s + 0.0511·35-s − 1.31·37-s − 0.160·39-s − 1.54·41-s − 0.794·43-s − 1.17·45-s + 0.671·47-s − 0.986·49-s + 1.80·51-s − 0.441·53-s − 0.715·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 0.302T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.51T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−7.14667009495707518574667036505, −6.72236717090708343800605199924, −6.32509308048312561714142557207, −5.35342812878046297551027278564, −4.94001319653524944707158482513, −4.01259858580927532142217898112, −3.55971082638454320585024823335, −1.84257208463999278687447181318, −1.04293745896348535530588738323, 0,
1.04293745896348535530588738323, 1.84257208463999278687447181318, 3.55971082638454320585024823335, 4.01259858580927532142217898112, 4.94001319653524944707158482513, 5.35342812878046297551027278564, 6.32509308048312561714142557207, 6.72236717090708343800605199924, 7.14667009495707518574667036505