L(s) = 1 | + 5-s − 13-s − 2·17-s − 2·19-s − 6·23-s + 25-s + 2·29-s − 6·31-s + 6·37-s − 10·41-s − 4·43-s + 12·53-s − 12·59-s − 65-s + 16·67-s − 8·71-s + 14·73-s − 4·83-s − 2·85-s + 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.64·53-s − 1.56·59-s − 0.124·65-s + 1.95·67-s − 0.949·71-s + 1.63·73-s − 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.205·95-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.12673290952895, −12.82157332118829, −12.14233347320047, −11.87370377117376, −11.32649489939382, −10.72265862578383, −10.45114963394925, −9.767181579948391, −9.620505593559337, −8.876710512664346, −8.541427143106358, −7.985729878080452, −7.530874741067508, −6.870919472509961, −6.499313816031756, −6.032011277862748, −5.416746456555699, −5.024301017846202, −4.370865412567449, −3.894319582943913, −3.325837860172348, −2.609887967384807, −2.052944947027862, −1.679623310474213, −0.7330823171911563, 0,
0.7330823171911563, 1.679623310474213, 2.052944947027862, 2.609887967384807, 3.325837860172348, 3.894319582943913, 4.370865412567449, 5.024301017846202, 5.416746456555699, 6.032011277862748, 6.499313816031756, 6.870919472509961, 7.530874741067508, 7.985729878080452, 8.541427143106358, 8.876710512664346, 9.620505593559337, 9.767181579948391, 10.45114963394925, 10.72265862578383, 11.32649489939382, 11.87370377117376, 12.14233347320047, 12.82157332118829, 13.12673290952895