L(s) = 1 | − 2.23·2-s − 0.381·3-s + 3.00·4-s + 0.854·6-s − 1.23·7-s − 2.23·8-s − 2.85·9-s − 1.61·11-s − 1.14·12-s − 2.85·13-s + 2.76·14-s − 0.999·16-s + 5.09·17-s + 6.38·18-s − 5.38·19-s + 0.472·21-s + 3.61·22-s + 4.47·23-s + 0.854·24-s + 6.38·26-s + 2.23·27-s − 3.70·28-s − 7.09·29-s + 2.47·31-s + 6.70·32-s + 0.618·33-s − 11.3·34-s + ⋯ |
L(s) = 1 | − 1.58·2-s − 0.220·3-s + 1.50·4-s + 0.348·6-s − 0.467·7-s − 0.790·8-s − 0.951·9-s − 0.487·11-s − 0.330·12-s − 0.791·13-s + 0.738·14-s − 0.249·16-s + 1.23·17-s + 1.50·18-s − 1.23·19-s + 0.103·21-s + 0.771·22-s + 0.932·23-s + 0.174·24-s + 1.25·26-s + 0.430·27-s − 0.700·28-s − 1.31·29-s + 0.444·31-s + 1.18·32-s + 0.107·33-s − 1.95·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 3.38T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 0.909T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 7.23T + 97T^{2} \) |
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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
−7.990826264581264827518262314735, −7.16806921806607797838383517961, −6.60626158602514357383050825672, −5.70514619004060113225561410790, −5.09035517761714538665271601876, −3.89609400861748581854196560595, −2.79343411246067938360707577060, −2.22941467887679852569963123646, −0.906722131916748905367403451711, 0,
0.906722131916748905367403451711, 2.22941467887679852569963123646, 2.79343411246067938360707577060, 3.89609400861748581854196560595, 5.09035517761714538665271601876, 5.70514619004060113225561410790, 6.60626158602514357383050825672, 7.16806921806607797838383517961, 7.990826264581264827518262314735