| L(s) = 1 | − 3-s − 5-s + 3.12·7-s + 9-s + 11-s + 5.12·13-s + 15-s − 5.12·17-s − 3.12·21-s − 4·23-s + 25-s − 27-s + 2·29-s − 33-s − 3.12·35-s + 6·37-s − 5.12·39-s + 6·41-s − 3.12·43-s − 45-s + 4·47-s + 2.75·49-s + 5.12·51-s + 8.24·53-s − 55-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.18·7-s + 0.333·9-s + 0.301·11-s + 1.42·13-s + 0.258·15-s − 1.24·17-s − 0.681·21-s − 0.834·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.174·33-s − 0.527·35-s + 0.986·37-s − 0.820·39-s + 0.937·41-s − 0.476·43-s − 0.149·45-s + 0.583·47-s + 0.393·49-s + 0.717·51-s + 1.13·53-s − 0.134·55-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505130431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505130431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.660453784589624343922139692012, −8.554052647885741308328579466048, −8.213274300683371859981225652762, −7.12290540691687084793027189216, −6.31302706333399249345648539357, −5.45471202853487126074755065994, −4.41954484398844085053880964208, −3.86572890316638213149104375862, −2.20441821729702906718946433058, −0.976174657239473000499428641341,
0.976174657239473000499428641341, 2.20441821729702906718946433058, 3.86572890316638213149104375862, 4.41954484398844085053880964208, 5.45471202853487126074755065994, 6.31302706333399249345648539357, 7.12290540691687084793027189216, 8.213274300683371859981225652762, 8.554052647885741308328579466048, 9.660453784589624343922139692012
