| L(s) = 1 | − 7-s + 2·17-s + 4·19-s + 2·23-s − 5·25-s − 4·29-s − 6·37-s + 4·41-s + 4·43-s + 6·47-s + 49-s + 4·53-s + 10·59-s − 10·61-s − 8·67-s + 8·71-s − 2·73-s − 8·79-s + 2·83-s + 16·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.485·17-s + 0.917·19-s + 0.417·23-s − 25-s − 0.742·29-s − 0.986·37-s + 0.624·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s + 0.549·53-s + 1.30·59-s − 1.28·61-s − 0.977·67-s + 0.949·71-s − 0.234·73-s − 0.900·79-s + 0.219·83-s + 1.69·89-s − 1.42·97-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.10589151458281, −13.65441695632966, −13.27016089034563, −12.67385808261479, −12.13410138377224, −11.81961008515940, −11.21294995331642, −10.65966553993918, −10.20079995819915, −9.615842747297795, −9.236647561070275, −8.738629795729041, −8.048369559690210, −7.426463544397097, −7.263594247781649, −6.451708992676791, −5.882512298491597, −5.437519034966693, −4.926299237871089, −4.014061441889167, −3.728957397641033, −2.979232527250014, −2.421203552167317, −1.602113463119695, −0.9043197638511676, 0,
0.9043197638511676, 1.602113463119695, 2.421203552167317, 2.979232527250014, 3.728957397641033, 4.014061441889167, 4.926299237871089, 5.437519034966693, 5.882512298491597, 6.451708992676791, 7.263594247781649, 7.426463544397097, 8.048369559690210, 8.738629795729041, 9.236647561070275, 9.615842747297795, 10.20079995819915, 10.65966553993918, 11.21294995331642, 11.81961008515940, 12.13410138377224, 12.67385808261479, 13.27016089034563, 13.65441695632966, 14.10589151458281
