L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s − 4·11-s − 2·12-s + 3·13-s − 15-s + 4·16-s + 17-s + 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 3·29-s − 6·31-s − 4·33-s − 35-s − 2·36-s + 6·37-s + 3·39-s + 12·41-s − 6·43-s + 8·44-s − 45-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.832·13-s − 0.258·15-s + 16-s + 0.242·17-s + 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.557·29-s − 1.07·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s + 0.986·37-s + 0.480·39-s + 1.87·41-s − 0.914·43-s + 1.20·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690841743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690841743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.41002510537340, −13.73953394194163, −13.54697206442297, −12.86850684037288, −12.59932624469724, −12.00726510891917, −11.15447122506921, −10.74203920916458, −10.43091839156666, −9.418307359817429, −9.327523701848946, −8.706656061695909, −7.972188077746858, −7.836096751793204, −7.395397937646047, −6.386787293953621, −5.737274866663499, −5.265149991810291, −4.570783819620774, −4.123716380207762, −3.489887429282206, −2.946574395718199, −2.137687326361583, −1.271862350640893, −0.4717942214636832,
0.4717942214636832, 1.271862350640893, 2.137687326361583, 2.946574395718199, 3.489887429282206, 4.123716380207762, 4.570783819620774, 5.265149991810291, 5.737274866663499, 6.386787293953621, 7.395397937646047, 7.836096751793204, 7.972188077746858, 8.706656061695909, 9.327523701848946, 9.418307359817429, 10.43091839156666, 10.74203920916458, 11.15447122506921, 12.00726510891917, 12.59932624469724, 12.86850684037288, 13.54697206442297, 13.73953394194163, 14.41002510537340