L(s) = 1 | + 2-s + 3.23·3-s + 4-s − 3.23·5-s + 3.23·6-s + 8-s + 7.47·9-s − 3.23·10-s + 11-s + 3.23·12-s − 1.23·13-s − 10.4·15-s + 16-s + 6.47·17-s + 7.47·18-s + 2.76·19-s − 3.23·20-s + 22-s + 4·23-s + 3.23·24-s + 5.47·25-s − 1.23·26-s + 14.4·27-s − 4.47·29-s − 10.4·30-s − 2·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.86·3-s + 0.5·4-s − 1.44·5-s + 1.32·6-s + 0.353·8-s + 2.49·9-s − 1.02·10-s + 0.301·11-s + 0.934·12-s − 0.342·13-s − 2.70·15-s + 0.250·16-s + 1.56·17-s + 1.76·18-s + 0.634·19-s − 0.723·20-s + 0.213·22-s + 0.834·23-s + 0.660·24-s + 1.09·25-s − 0.242·26-s + 2.78·27-s − 0.830·29-s − 1.91·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.939728353\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.939728353\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−9.743913546027839964224405485591, −8.868440804206357708140504609090, −8.107011218752175823817849183982, −7.42804411366250484970207537346, −7.03000964499145678562181388468, −5.28717239674767381762556781599, −4.24663132650566531859521090082, −3.43146858905407759239798601585, −3.08791371561946288993837862507, −1.55719080927706853210503358377,
1.55719080927706853210503358377, 3.08791371561946288993837862507, 3.43146858905407759239798601585, 4.24663132650566531859521090082, 5.28717239674767381762556781599, 7.03000964499145678562181388468, 7.42804411366250484970207537346, 8.107011218752175823817849183982, 8.868440804206357708140504609090, 9.743913546027839964224405485591