| L(s) = 1 | + 2·5-s − 2·7-s − 6·11-s − 13-s + 2·17-s + 6·19-s − 25-s + 6·29-s − 6·31-s − 4·35-s + 2·37-s + 10·41-s − 8·43-s + 6·47-s − 3·49-s − 6·53-s − 12·55-s − 6·59-s − 10·61-s − 2·65-s − 2·67-s − 14·71-s − 14·73-s + 12·77-s − 4·79-s + 6·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.80·11-s − 0.277·13-s + 0.485·17-s + 1.37·19-s − 1/5·25-s + 1.11·29-s − 1.07·31-s − 0.676·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.61·55-s − 0.781·59-s − 1.28·61-s − 0.248·65-s − 0.244·67-s − 1.66·71-s − 1.63·73-s + 1.36·77-s − 0.450·79-s + 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.915048911369435591002695098729, −7.56957107687067026532497329945, −6.61038127680129135994910392337, −5.68813498429657860376688452894, −5.41819068226409293384730615526, −4.42205609107136233613776718732, −3.05523020149450579500366698279, −2.74710804091032358687905405889, −1.49159930001743979277115584087, 0,
1.49159930001743979277115584087, 2.74710804091032358687905405889, 3.05523020149450579500366698279, 4.42205609107136233613776718732, 5.41819068226409293384730615526, 5.68813498429657860376688452894, 6.61038127680129135994910392337, 7.56957107687067026532497329945, 7.915048911369435591002695098729
