L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s − 2·9-s − 10-s − 12-s + 6·13-s + 3·14-s − 15-s + 16-s + 7·17-s + 2·18-s − 5·19-s + 20-s + 3·21-s − 6·23-s + 24-s + 25-s − 6·26-s + 5·27-s − 3·28-s − 5·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.69·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s + 0.654·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.962·27-s − 0.566·28-s − 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.407733303030992428027906608319, −8.581836953357011865271355123460, −7.85845755176844646843682904510, −6.59397912005552108069880429500, −6.06279392030344526790826351560, −5.55644984550596935545377572587, −3.84267487580452379681647251384, −2.98710311935352308879373204244, −1.51256845978893665060018342495, 0,
1.51256845978893665060018342495, 2.98710311935352308879373204244, 3.84267487580452379681647251384, 5.55644984550596935545377572587, 6.06279392030344526790826351560, 6.59397912005552108069880429500, 7.85845755176844646843682904510, 8.581836953357011865271355123460, 9.407733303030992428027906608319