| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s + 5·11-s + 2·13-s − 15-s − 17-s − 19-s − 3·21-s + 4·23-s − 4·25-s + 27-s + 6·29-s − 10·31-s + 5·33-s + 3·35-s + 2·39-s + 11·43-s − 45-s + 9·47-s + 2·49-s − 51-s − 10·53-s − 5·55-s − 57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.554·13-s − 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s + 0.870·33-s + 0.507·35-s + 0.320·39-s + 1.67·43-s − 0.149·45-s + 1.31·47-s + 2/7·49-s − 0.140·51-s − 1.37·53-s − 0.674·55-s − 0.132·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.029901592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.029901592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.684109523738480154200581620607, −7.81487350966492974578323122788, −6.97162896330102730850752382154, −6.49289150756279373767286826955, −5.70455068044459736154986183694, −4.42131885288991939036937111558, −3.75730418962833054514652767128, −3.22909189689445240832756166916, −2.06480175292692521017322664960, −0.820320077442689800183422611185,
0.820320077442689800183422611185, 2.06480175292692521017322664960, 3.22909189689445240832756166916, 3.75730418962833054514652767128, 4.42131885288991939036937111558, 5.70455068044459736154986183694, 6.49289150756279373767286826955, 6.97162896330102730850752382154, 7.81487350966492974578323122788, 8.684109523738480154200581620607
