| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 7-s − 8-s + 9-s + 10-s − 11-s + 2·12-s − 4·13-s − 14-s − 2·15-s + 16-s − 3·17-s − 18-s − 20-s + 2·21-s + 22-s − 4·23-s − 2·24-s + 25-s + 4·26-s − 4·27-s + 28-s − 5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.79616771503206, −12.70466959170680, −11.93468487531917, −11.58916385181963, −11.06174892989098, −10.70831667591357, −10.06906288187967, −9.592708956560817, −9.277929690829161, −8.788526765073321, −8.352966561202303, −7.821445437990395, −7.661601053970449, −7.097421617316020, −6.686267432841280, −5.821510384540116, −5.469725217668002, −4.767114978916465, −4.167171055477833, −3.723794925494460, −3.114345824825577, −2.513360466492896, −2.096831956312336, −1.714188036306064, −0.6377458070626911, 0,
0.6377458070626911, 1.714188036306064, 2.096831956312336, 2.513360466492896, 3.114345824825577, 3.723794925494460, 4.167171055477833, 4.767114978916465, 5.469725217668002, 5.821510384540116, 6.686267432841280, 7.097421617316020, 7.661601053970449, 7.821445437990395, 8.352966561202303, 8.788526765073321, 9.277929690829161, 9.592708956560817, 10.06906288187967, 10.70831667591357, 11.06174892989098, 11.58916385181963, 11.93468487531917, 12.70466959170680, 12.79616771503206
