| L(s) = 1 | + 0.709·2-s − 1.77·3-s − 0.497·4-s − 5-s − 1.25·6-s − 0.241·7-s − 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s + 1.49·17-s + 1.51·18-s + 0.497·20-s + 0.426·21-s − 1.37·22-s − 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s − 2.01·27-s + 0.119·28-s + 1.25·30-s + ⋯ |
| L(s) = 1 | + 0.709·2-s − 1.77·3-s − 0.497·4-s − 5-s − 1.25·6-s − 0.241·7-s − 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s + 0.880·12-s − 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s + 1.49·17-s + 1.51·18-s + 0.497·20-s + 0.426·21-s − 1.37·22-s − 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s − 2.01·27-s + 0.119·28-s + 1.25·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3283561833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3283561833\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.709T + T^{2} \) |
| 3 | \( 1 + 1.77T + T^{2} \) |
| 7 | \( 1 + 0.241T + T^{2} \) |
| 11 | \( 1 + 1.94T + T^{2} \) |
| 17 | \( 1 - 1.49T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.13T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.709T + T^{2} \) |
| 47 | \( 1 + 1.13T + T^{2} \) |
| 53 | \( 1 - 1.94T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.94T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.241T + T^{2} \) |
| 97 | \( 1 + 1.77T + T^{2} \) |
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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
−9.872337804061178285541654464174, −8.221252497897020497670703862246, −7.71796335558993246889157565980, −6.80699318287326320739260551755, −5.79365521696413465318947529852, −5.25646021643500113343884938477, −4.75446753065031788836569066579, −3.87673037531629597240929452920, −2.77709517962760665606276343950, −0.52864828207844568732104807898,
0.52864828207844568732104807898, 2.77709517962760665606276343950, 3.87673037531629597240929452920, 4.75446753065031788836569066579, 5.25646021643500113343884938477, 5.79365521696413465318947529852, 6.80699318287326320739260551755, 7.71796335558993246889157565980, 8.221252497897020497670703862246, 9.872337804061178285541654464174
