| L(s) = 1 | + 3.31·2-s + 7·4-s − 5·5-s − 13.2·7-s + 9.94·8-s + 9·9-s − 16.5·10-s + 11·11-s + 13.2·13-s − 44·14-s + 5.00·16-s − 13.2·17-s + 29.8·18-s − 35·20-s + 36.4·22-s + 25·25-s + 44·26-s − 92.8·28-s − 18·31-s − 23.2·32-s − 44·34-s + 66.3·35-s + 63·36-s − 49.7·40-s + 13.2·43-s + 77·44-s − 45·45-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.75·4-s − 5-s − 1.89·7-s + 1.24·8-s + 9-s − 1.65·10-s + 11-s + 1.02·13-s − 3.14·14-s + 0.312·16-s − 0.780·17-s + 1.65·18-s − 1.75·20-s + 1.65·22-s + 25-s + 1.69·26-s − 3.31·28-s − 0.580·31-s − 0.725·32-s − 1.29·34-s + 1.89·35-s + 1.75·36-s − 1.24·40-s + 0.308·43-s + 1.75·44-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.160980100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.160980100\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.31T + 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 + 13.2T + 49T^{2} \) |
| 13 | \( 1 - 13.2T + 169T^{2} \) |
| 17 | \( 1 + 13.2T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 18T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 102T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 78T + 5.04e3T^{2} \) |
| 73 | \( 1 - 145.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−15.15977275342733538936259310541, −13.68604631089774728647444598586, −12.85751225428424224926627399519, −12.16086094783261546702091020396, −10.89069731795278505466154426713, −9.216808936737792909745846887712, −7.01006237305235694954774519466, −6.26572071138714327474507278724, −4.20455777884498723889584370787, −3.41520667080129417067827701771,
3.41520667080129417067827701771, 4.20455777884498723889584370787, 6.26572071138714327474507278724, 7.01006237305235694954774519466, 9.216808936737792909745846887712, 10.89069731795278505466154426713, 12.16086094783261546702091020396, 12.85751225428424224926627399519, 13.68604631089774728647444598586, 15.15977275342733538936259310541
