| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 7-s + 8-s + 9-s + 2·10-s + 4·11-s − 12-s − 6·13-s − 14-s − 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 8·23-s − 24-s − 25-s − 6·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92778 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92778 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 47 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.02464299334945, −13.83248451647171, −13.01951659822444, −12.50773717237814, −12.13047378917093, −11.80804943848531, −11.39505280844051, −10.47098300331857, −10.12401692913173, −9.741181553975858, −9.362351062878588, −8.665427433809357, −7.812466344074141, −7.347168106133333, −6.823114730118541, −6.393749211897422, −5.634661791728373, −5.583491625811294, −4.879692401643342, −4.143095342946685, −3.800666547929241, −2.944073798444713, −2.317639243554027, −1.750683931081212, −1.022118967082164, 0,
1.022118967082164, 1.750683931081212, 2.317639243554027, 2.944073798444713, 3.800666547929241, 4.143095342946685, 4.879692401643342, 5.583491625811294, 5.634661791728373, 6.393749211897422, 6.823114730118541, 7.347168106133333, 7.812466344074141, 8.665427433809357, 9.362351062878588, 9.741181553975858, 10.12401692913173, 10.47098300331857, 11.39505280844051, 11.80804943848531, 12.13047378917093, 12.50773717237814, 13.01951659822444, 13.83248451647171, 14.02464299334945
