| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 6·13-s − 14-s + 16-s − 2·17-s + 20-s − 22-s + 25-s + 6·26-s − 28-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 35-s + 10·37-s + 40-s + 2·41-s − 8·43-s − 44-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 0.213·22-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 1.64·37-s + 0.158·40-s + 0.312·41-s − 1.21·43-s − 0.150·44-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.712717135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.712717135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.040807870107218540618215960715, −6.88341246292538720939166603228, −6.56743589912726554451228933377, −5.76091587192600836600482271945, −5.27525787823813355786529713765, −4.25214485336933234982916082566, −3.66434120439946026424503140636, −2.82512222904602526635901782881, −1.97294971152287585129887321522, −0.905693622107952988773167673539,
0.905693622107952988773167673539, 1.97294971152287585129887321522, 2.82512222904602526635901782881, 3.66434120439946026424503140636, 4.25214485336933234982916082566, 5.27525787823813355786529713765, 5.76091587192600836600482271945, 6.56743589912726554451228933377, 6.88341246292538720939166603228, 8.040807870107218540618215960715
