| L(s) = 1 | + 2·11-s + 13-s − 6·17-s + 4·19-s − 4·23-s − 5·25-s − 10·29-s + 8·31-s − 2·37-s − 4·43-s + 2·47-s + 2·53-s + 10·59-s − 10·61-s + 8·67-s − 2·71-s + 10·73-s + 8·79-s + 6·83-s − 12·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s − 25-s − 1.85·29-s + 1.43·31-s − 0.328·37-s − 0.609·43-s + 0.291·47-s + 0.274·53-s + 1.30·59-s − 1.28·61-s + 0.977·67-s − 0.237·71-s + 1.17·73-s + 0.900·79-s + 0.658·83-s − 1.27·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.97159139342418, −14.27406818150711, −13.67086836852627, −13.57706048885360, −12.87003232975112, −12.23385534166068, −11.69480695011820, −11.34419387060851, −10.85822209276726, −10.05490011482517, −9.673219615989826, −9.156850351353414, −8.549971204535781, −8.070255934430190, −7.368642769481067, −6.921226857439713, −6.180121781822077, −5.870228192970721, −5.046990518632616, −4.460188409737115, −3.775596727057188, −3.397811362169130, −2.289340240890687, −1.927847454530567, −0.9538840520714931, 0,
0.9538840520714931, 1.927847454530567, 2.289340240890687, 3.397811362169130, 3.775596727057188, 4.460188409737115, 5.046990518632616, 5.870228192970721, 6.180121781822077, 6.921226857439713, 7.368642769481067, 8.070255934430190, 8.549971204535781, 9.156850351353414, 9.673219615989826, 10.05490011482517, 10.85822209276726, 11.34419387060851, 11.69480695011820, 12.23385534166068, 12.87003232975112, 13.57706048885360, 13.67086836852627, 14.27406818150711, 14.97159139342418
