| L(s) = 1 | − 3.25·3-s + 7.62·9-s + 11-s − 5.29·13-s − 2.03·17-s + 4.48·19-s − 7.62·23-s − 15.0·27-s − 5.62·29-s + 1.22·31-s − 3.25·33-s + 7.62·37-s + 17.2·39-s − 9.77·41-s + 11.6·43-s − 6.51·47-s + 6.62·51-s − 11.2·53-s − 14.6·57-s + 5.29·59-s − 7.74·61-s + 8.24·67-s + 24.8·69-s + 2.37·71-s + 16.2·73-s + 5.62·79-s + 26.2·81-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 2.54·9-s + 0.301·11-s − 1.46·13-s − 0.492·17-s + 1.02·19-s − 1.58·23-s − 2.90·27-s − 1.04·29-s + 0.220·31-s − 0.567·33-s + 1.25·37-s + 2.76·39-s − 1.52·41-s + 1.77·43-s − 0.950·47-s + 0.927·51-s − 1.54·53-s − 1.93·57-s + 0.688·59-s − 0.991·61-s + 1.00·67-s + 2.99·69-s + 0.282·71-s + 1.90·73-s + 0.632·79-s + 2.91·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5044553774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5044553774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 5.62T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 - 7.62T + 37T^{2} \) |
| 41 | \( 1 + 9.77T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 0.804T + 83T^{2} \) |
| 89 | \( 1 - 0.804T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.58638899824764008834214831784, −6.74651898876816170306677413099, −6.34269096812545887019948208107, −5.54101369699747572966027452940, −5.08656020630590557996752805308, −4.41865405007240914182988856263, −3.72429811257568522741361783376, −2.39248770770020667438062764603, −1.48721739684222064034241394804, −0.38414409146716909912625262272,
0.38414409146716909912625262272, 1.48721739684222064034241394804, 2.39248770770020667438062764603, 3.72429811257568522741361783376, 4.41865405007240914182988856263, 5.08656020630590557996752805308, 5.54101369699747572966027452940, 6.34269096812545887019948208107, 6.74651898876816170306677413099, 7.58638899824764008834214831784
