| L(s) = 1 | − 3-s − 3.41·5-s − 0.828·7-s + 9-s − 11-s − 13-s + 3.41·15-s − 2.58·17-s + 6·19-s + 0.828·21-s − 4.82·23-s + 6.65·25-s − 27-s − 4.24·29-s + 4.24·31-s + 33-s + 2.82·35-s − 3.65·37-s + 39-s − 12·41-s + 9.07·43-s − 3.41·45-s − 8.48·47-s − 6.31·49-s + 2.58·51-s − 9.31·53-s + 3.41·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s − 0.313·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.881·15-s − 0.627·17-s + 1.37·19-s + 0.180·21-s − 1.00·23-s + 1.33·25-s − 0.192·27-s − 0.787·29-s + 0.762·31-s + 0.174·33-s + 0.478·35-s − 0.601·37-s + 0.160·39-s − 1.87·41-s + 1.38·43-s − 0.508·45-s − 1.23·47-s − 0.901·49-s + 0.362·51-s − 1.27·53-s + 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4129378479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4129378479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.84127199644200803477733710914, −7.33331351379144109626436111544, −6.67358052099800563034324488978, −5.86476542909370202665499993979, −4.98870268667790047281828633718, −4.43745493738312617251053801365, −3.59225977071731768795709619725, −2.99923986550735956405700748600, −1.65687013718472957407199557059, −0.33529981501948817751693253191,
0.33529981501948817751693253191, 1.65687013718472957407199557059, 2.99923986550735956405700748600, 3.59225977071731768795709619725, 4.43745493738312617251053801365, 4.98870268667790047281828633718, 5.86476542909370202665499993979, 6.67358052099800563034324488978, 7.33331351379144109626436111544, 7.84127199644200803477733710914
