| L(s) = 1 | − 6·11-s + 13-s − 4·17-s + 6·19-s − 6·23-s + 2·29-s − 6·37-s − 2·41-s − 12·43-s − 12·47-s − 7·49-s − 10·53-s − 6·59-s + 10·61-s − 12·67-s − 12·71-s + 2·73-s − 12·83-s + 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 0.277·13-s − 0.970·17-s + 1.37·19-s − 1.25·23-s + 0.371·29-s − 0.986·37-s − 0.312·41-s − 1.82·43-s − 1.75·47-s − 49-s − 1.37·53-s − 0.781·59-s + 1.28·61-s − 1.46·67-s − 1.42·71-s + 0.234·73-s − 1.31·83-s + 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.19257066264422, −13.72043148153779, −13.31364902266985, −13.03273504819600, −12.32530424481413, −11.87756238501135, −11.29347096710358, −10.95990033317442, −10.16597204276542, −10.03537666632828, −9.495547673974185, −8.613398786822839, −8.370117156248511, −7.763517691245659, −7.417055665135102, −6.612408812528132, −6.269454499852184, −5.465743692678418, −5.065927606948558, −4.665425504943184, −3.833237400721217, −3.096247952137931, −2.838670610873839, −1.880805221667760, −1.454934498983247, 0, 0,
1.454934498983247, 1.880805221667760, 2.838670610873839, 3.096247952137931, 3.833237400721217, 4.665425504943184, 5.065927606948558, 5.465743692678418, 6.269454499852184, 6.612408812528132, 7.417055665135102, 7.763517691245659, 8.370117156248511, 8.613398786822839, 9.495547673974185, 10.03537666632828, 10.16597204276542, 10.95990033317442, 11.29347096710358, 11.87756238501135, 12.32530424481413, 13.03273504819600, 13.31364902266985, 13.72043148153779, 14.19257066264422
