| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 11-s − 12-s − 6·13-s + 4·14-s + 16-s − 17-s − 18-s − 4·19-s + 4·21-s + 22-s − 4·23-s + 24-s + 6·26-s − 27-s − 4·28-s − 6·29-s + 4·31-s − 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.872·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.66692972012041, −15.12612228728796, −14.60681250970525, −13.82865276353141, −13.17366199222411, −12.56401432682143, −12.45760090107155, −11.66513903783947, −11.21389730397554, −10.32103969495135, −10.07202534919887, −9.734226787944211, −9.038704373237256, −8.475232932542633, −7.608378793954353, −7.224601753792444, −6.612039515592281, −6.143420481575591, −5.518216645425029, −4.779942186477656, −4.016398901679501, −3.279022602068504, −2.444097033829544, −1.993591598892661, −0.5926079736986398, 0,
0.5926079736986398, 1.993591598892661, 2.444097033829544, 3.279022602068504, 4.016398901679501, 4.779942186477656, 5.518216645425029, 6.143420481575591, 6.612039515592281, 7.224601753792444, 7.608378793954353, 8.475232932542633, 9.038704373237256, 9.734226787944211, 10.07202534919887, 10.32103969495135, 11.21389730397554, 11.66513903783947, 12.45760090107155, 12.56401432682143, 13.17366199222411, 13.82865276353141, 14.60681250970525, 15.12612228728796, 15.66692972012041
