| L(s) = 1 | − 3-s − 2.37·5-s − 7-s + 9-s − 2·11-s − 13-s + 2.37·15-s − 6.37·19-s + 21-s − 4.37·23-s + 0.627·25-s − 27-s − 4.37·29-s − 2.37·31-s + 2·33-s + 2.37·35-s + 6.74·37-s + 39-s + 8.74·41-s − 11.1·43-s − 2.37·45-s + 6.37·47-s + 49-s − 0.372·53-s + 4.74·55-s + 6.37·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.06·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s − 0.277·13-s + 0.612·15-s − 1.46·19-s + 0.218·21-s − 0.911·23-s + 0.125·25-s − 0.192·27-s − 0.811·29-s − 0.426·31-s + 0.348·33-s + 0.400·35-s + 1.10·37-s + 0.160·39-s + 1.36·41-s − 1.69·43-s − 0.353·45-s + 0.929·47-s + 0.142·49-s − 0.0511·53-s + 0.639·55-s + 0.844·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4370465317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4370465317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 + 0.372T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 2.74T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.249473605685829877364603480190, −7.59357320273006210296343134822, −7.01811843996224926472045534946, −6.09190148526156743915440064587, −5.54123113914894278657470887588, −4.36339911723814671209082687899, −4.09777281569580816708423624964, −2.97842774283939190974358528708, −1.94224996286917475886429854434, −0.36409580788311828761240336846,
0.36409580788311828761240336846, 1.94224996286917475886429854434, 2.97842774283939190974358528708, 4.09777281569580816708423624964, 4.36339911723814671209082687899, 5.54123113914894278657470887588, 6.09190148526156743915440064587, 7.01811843996224926472045534946, 7.59357320273006210296343134822, 8.249473605685829877364603480190
