| L(s) = 1 | − 2.51·3-s + 4.64·7-s + 3.32·9-s − 1.64·11-s − 1.91·13-s − 0.969·17-s − 4.91·19-s − 11.6·21-s − 23-s − 0.825·27-s + 7.48·29-s + 7.77·31-s + 4.14·33-s − 7.63·37-s + 4.82·39-s + 5.79·41-s + 3.77·43-s + 2.22·47-s + 14.5·49-s + 2.43·51-s − 11.2·53-s + 12.3·57-s + 10.1·59-s + 6.06·61-s + 15.4·63-s − 12.0·67-s + 2.51·69-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 1.75·7-s + 1.10·9-s − 0.497·11-s − 0.531·13-s − 0.235·17-s − 1.12·19-s − 2.54·21-s − 0.208·23-s − 0.158·27-s + 1.38·29-s + 1.39·31-s + 0.722·33-s − 1.25·37-s + 0.771·39-s + 0.905·41-s + 0.575·43-s + 0.324·47-s + 2.07·49-s + 0.341·51-s − 1.55·53-s + 1.63·57-s + 1.31·59-s + 0.776·61-s + 1.94·63-s − 1.47·67-s + 0.302·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171764684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171764684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.51T + 3T^{2} \) |
| 7 | \( 1 - 4.64T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 + 0.969T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 7.63T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 3.01T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 0.163T + 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−8.284926446240835991922770227382, −7.55553376699960986192658669066, −6.76774977035622724850446216135, −6.02780112225357024015056480721, −5.32246490459947940850365358550, −4.62096165219114767830700331872, −4.37628270545068775774651376739, −2.68857928723238295502231878067, −1.73272119099020750744875027655, −0.66357789960710625453594169679,
0.66357789960710625453594169679, 1.73272119099020750744875027655, 2.68857928723238295502231878067, 4.37628270545068775774651376739, 4.62096165219114767830700331872, 5.32246490459947940850365358550, 6.02780112225357024015056480721, 6.76774977035622724850446216135, 7.55553376699960986192658669066, 8.284926446240835991922770227382
