L(s) = 1 | − 0.875·3-s + 5-s − 1.42·7-s − 2.23·9-s + 2.99·11-s − 3.93·13-s − 0.875·15-s − 6.64·17-s − 7.14·19-s + 1.24·21-s + 6.98·23-s + 25-s + 4.58·27-s + 7.63·29-s − 1.79·31-s − 2.61·33-s − 1.42·35-s − 0.111·37-s + 3.44·39-s − 6.42·41-s − 7.93·43-s − 2.23·45-s − 6.10·47-s − 4.97·49-s + 5.81·51-s + 3.14·53-s + 2.99·55-s + ⋯ |
L(s) = 1 | − 0.505·3-s + 0.447·5-s − 0.537·7-s − 0.744·9-s + 0.901·11-s − 1.09·13-s − 0.226·15-s − 1.61·17-s − 1.63·19-s + 0.271·21-s + 1.45·23-s + 0.200·25-s + 0.882·27-s + 1.41·29-s − 0.323·31-s − 0.456·33-s − 0.240·35-s − 0.0183·37-s + 0.551·39-s − 1.00·41-s − 1.21·43-s − 0.332·45-s − 0.890·47-s − 0.711·49-s + 0.814·51-s + 0.432·53-s + 0.403·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9757758136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9757758136\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 0.875T + 3T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 7.14T + 19T^{2} \) |
| 23 | \( 1 - 6.98T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 + 0.111T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.391647759053864420185197531389, −6.86206967050230355738986561694, −6.71471703529383630457391324556, −6.17860992912145288315503734970, −5.00642227747897633282876012003, −4.75696275651030850893672736491, −3.61776855851947318375822928999, −2.67743744050059872470126724329, −1.95850342739226179391737379850, −0.50622259712963961140036019524,
0.50622259712963961140036019524, 1.95850342739226179391737379850, 2.67743744050059872470126724329, 3.61776855851947318375822928999, 4.75696275651030850893672736491, 5.00642227747897633282876012003, 6.17860992912145288315503734970, 6.71471703529383630457391324556, 6.86206967050230355738986561694, 8.391647759053864420185197531389