L(s) = 1 | + 5-s + 4·7-s − 4·13-s − 4·19-s + 8·23-s + 25-s − 4·29-s + 4·31-s + 4·35-s − 2·37-s + 4·41-s − 12·43-s + 8·47-s + 9·49-s + 10·53-s − 4·65-s − 4·67-s − 4·71-s + 4·73-s + 12·79-s + 16·83-s + 6·89-s − 16·91-s − 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.10·13-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s + 0.624·41-s − 1.82·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.496·65-s − 0.488·67-s − 0.474·71-s + 0.468·73-s + 1.35·79-s + 1.75·83-s + 0.635·89-s − 1.67·91-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.922683762\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.922683762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.11771380445879, −14.90864873914953, −14.79909738689647, −13.84686887728141, −13.51290075628321, −12.87159198115856, −12.15097876059424, −11.81324848067205, −11.06018257736820, −10.66375372089842, −10.16758212146482, −9.269824175552945, −8.978416183061579, −8.175489917845006, −7.788833705646726, −7.006890776908097, −6.612917648951859, −5.557801551471909, −5.166294970912398, −4.642257443198594, −3.987443138532659, −2.926368875112663, −2.247339398155893, −1.644173871126250, −0.7041258972448642,
0.7041258972448642, 1.644173871126250, 2.247339398155893, 2.926368875112663, 3.987443138532659, 4.642257443198594, 5.166294970912398, 5.557801551471909, 6.612917648951859, 7.006890776908097, 7.788833705646726, 8.175489917845006, 8.978416183061579, 9.269824175552945, 10.16758212146482, 10.66375372089842, 11.06018257736820, 11.81324848067205, 12.15097876059424, 12.87159198115856, 13.51290075628321, 13.84686887728141, 14.79909738689647, 14.90864873914953, 15.11771380445879