L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 3.46·13-s − 14-s + 16-s − 3.46·17-s + 20-s + 22-s − 5.46·23-s + 25-s + 3.46·26-s + 28-s − 2·29-s − 6.92·31-s − 32-s + 3.46·34-s + 35-s + 0.535·37-s − 40-s − 4.92·41-s + 10.9·43-s − 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.960·13-s − 0.267·14-s + 0.250·16-s − 0.840·17-s + 0.223·20-s + 0.213·22-s − 1.13·23-s + 0.200·25-s + 0.679·26-s + 0.188·28-s − 0.371·29-s − 1.24·31-s − 0.176·32-s + 0.594·34-s + 0.169·35-s + 0.0881·37-s − 0.158·40-s − 0.769·41-s + 1.66·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.164523443\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164523443\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.86817091426692951286676760735, −7.44159966918481628693241872822, −6.67645728759025185010571930130, −5.90624719315265733568864390077, −5.23563599171944562561386112902, −4.42277320494169648085506626055, −3.48121124671819618694571974185, −2.26015750238241324990905258330, −2.02277669445611681829690769554, −0.58675343213014481012918381257,
0.58675343213014481012918381257, 2.02277669445611681829690769554, 2.26015750238241324990905258330, 3.48121124671819618694571974185, 4.42277320494169648085506626055, 5.23563599171944562561386112902, 5.90624719315265733568864390077, 6.67645728759025185010571930130, 7.44159966918481628693241872822, 7.86817091426692951286676760735