| L(s) = 1 | + 2·7-s − 17-s + 6·19-s + 8·23-s − 5·25-s − 10·29-s + 10·31-s + 2·37-s − 4·41-s + 12·43-s − 12·47-s − 3·49-s + 2·53-s + 10·61-s − 6·67-s − 4·71-s − 10·73-s − 8·79-s − 12·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.242·17-s + 1.37·19-s + 1.66·23-s − 25-s − 1.85·29-s + 1.79·31-s + 0.328·37-s − 0.624·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.274·53-s + 1.28·61-s − 0.733·67-s − 0.474·71-s − 1.17·73-s − 0.900·79-s − 1.27·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.15401619273066, −13.36249012802750, −13.07113670495462, −12.63766641979197, −11.67167189964776, −11.52712136387246, −11.32457601935092, −10.51027966738228, −10.01246055965879, −9.525856466187179, −9.026887349449478, −8.523652757911670, −7.906146444072465, −7.448346736254589, −7.125532963400858, −6.317567769079730, −5.819504401137548, −5.193137957836686, −4.830445276915432, −4.190240642975473, −3.532985917305665, −2.927501274580201, −2.324443818309210, −1.470579670710227, −1.053996848182529, 0,
1.053996848182529, 1.470579670710227, 2.324443818309210, 2.927501274580201, 3.532985917305665, 4.190240642975473, 4.830445276915432, 5.193137957836686, 5.819504401137548, 6.317567769079730, 7.125532963400858, 7.448346736254589, 7.906146444072465, 8.523652757911670, 9.026887349449478, 9.525856466187179, 10.01246055965879, 10.51027966738228, 11.32457601935092, 11.52712136387246, 11.67167189964776, 12.63766641979197, 13.07113670495462, 13.36249012802750, 14.15401619273066
