| L(s) = 1 | + 4·7-s + 13-s + 3·19-s + 4·23-s + 29-s + 8·31-s + 37-s − 41-s + 6·43-s + 11·47-s + 9·49-s − 3·53-s + 10·59-s + 4·61-s + 13·67-s + 9·71-s + 3·79-s − 2·83-s − 10·89-s + 4·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.277·13-s + 0.688·19-s + 0.834·23-s + 0.185·29-s + 1.43·31-s + 0.164·37-s − 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s − 0.412·53-s + 1.30·59-s + 0.512·61-s + 1.58·67-s + 1.06·71-s + 0.337·79-s − 0.219·83-s − 1.05·89-s + 0.419·91-s + 0.812·97-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
| \(L(1)\) |
\(\approx\) |
\(4.550070776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.550070776\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.94233558269590, −13.47662891893998, −12.77299266970862, −12.35064982894474, −11.72803874718559, −11.34630629094900, −11.02641608634684, −10.40389572478739, −9.931130160270584, −9.272864922501530, −8.772695003126832, −8.224063663917346, −7.930993468902883, −7.264469253583800, −6.836985672310381, −6.122909067078706, −5.445136086354128, −5.119542213444373, −4.488803738397935, −4.026226431385478, −3.273583833646157, −2.520194711914288, −2.041322828387786, −1.068604367936621, −0.8399980321543461,
0.8399980321543461, 1.068604367936621, 2.041322828387786, 2.520194711914288, 3.273583833646157, 4.026226431385478, 4.488803738397935, 5.119542213444373, 5.445136086354128, 6.122909067078706, 6.836985672310381, 7.264469253583800, 7.930993468902883, 8.224063663917346, 8.772695003126832, 9.272864922501530, 9.931130160270584, 10.40389572478739, 11.02641608634684, 11.34630629094900, 11.72803874718559, 12.35064982894474, 12.77299266970862, 13.47662891893998, 13.94233558269590
