| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s − 2·15-s + 4·17-s + 4·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 10·29-s + 4·31-s − 2·33-s + 4·35-s − 2·37-s − 8·39-s − 10·41-s + 8·43-s − 45-s − 6·47-s + 9·49-s + 8·51-s − 2·53-s + 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.516·15-s + 0.970·17-s + 0.917·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s + 0.718·31-s − 0.348·33-s + 0.676·35-s − 0.328·37-s − 1.28·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 1.12·51-s − 0.274·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930006648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930006648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.527521801955114252432580447305, −7.923580520667090509256462573465, −7.14162936849208426215495288495, −6.62536331191924721842679282461, −5.48691634933986413556576113146, −4.69737474865266608096337291625, −3.42894015870646694064785929471, −3.15044879752992676393154963603, −2.43460898771104914069889479972, −0.75069098874977159462101565047,
0.75069098874977159462101565047, 2.43460898771104914069889479972, 3.15044879752992676393154963603, 3.42894015870646694064785929471, 4.69737474865266608096337291625, 5.48691634933986413556576113146, 6.62536331191924721842679282461, 7.14162936849208426215495288495, 7.923580520667090509256462573465, 8.527521801955114252432580447305
