| L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s + 9-s + 2·12-s + 15-s + 4·16-s − 17-s − 2·19-s + 2·20-s + 21-s − 9·23-s + 25-s − 27-s + 2·28-s − 10·29-s − 8·31-s + 35-s − 2·36-s − 9·37-s − 2·41-s + 2·43-s − 45-s + 4·47-s − 4·48-s + 49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.258·15-s + 16-s − 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.218·21-s − 1.87·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.85·29-s − 1.43·31-s + 0.169·35-s − 1/3·36-s − 1.47·37-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 0.583·47-s − 0.577·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.85319227822859, −12.43075460886245, −12.13749877035282, −11.57652983548079, −11.06314684836612, −10.56999574474335, −10.24456097364557, −9.648940963352063, −9.243772029008984, −8.858658766862124, −8.308511037250904, −7.769686978604100, −7.401928517238544, −6.881844508135937, −6.125682629702856, −5.890932975353656, −5.270613464954665, −4.925694211363622, −4.136381431607419, −3.855913586000904, −3.563926428138021, −2.698894618311590, −1.834676821637582, −1.489144563110116, −0.3421804019051265, 0,
0.3421804019051265, 1.489144563110116, 1.834676821637582, 2.698894618311590, 3.563926428138021, 3.855913586000904, 4.136381431607419, 4.925694211363622, 5.270613464954665, 5.890932975353656, 6.125682629702856, 6.881844508135937, 7.401928517238544, 7.769686978604100, 8.308511037250904, 8.858658766862124, 9.243772029008984, 9.648940963352063, 10.24456097364557, 10.56999574474335, 11.06314684836612, 11.57652983548079, 12.13749877035282, 12.43075460886245, 12.85319227822859
