| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·11-s + 13-s − 15-s + 17-s + 2·19-s + 2·21-s − 2·23-s + 25-s − 27-s + 29-s + 5·31-s − 2·33-s − 2·35-s − 39-s − 10·41-s − 6·43-s + 45-s − 8·47-s − 3·49-s − 51-s + 5·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.898·31-s − 0.348·33-s − 0.338·35-s − 0.160·39-s − 1.56·41-s − 0.914·43-s + 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.140·51-s + 0.686·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.15322626513276, −12.90716228302175, −12.14613477130651, −11.76767900397470, −11.65104421951223, −10.84275418354196, −10.28767615520016, −10.04214626197160, −9.640556877432695, −8.990858556629126, −8.678934907699866, −7.957955952283416, −7.507348663314552, −6.803292188678253, −6.428170193989252, −6.179441726875300, −5.562700226737648, −4.948361008288904, −4.596671501232553, −3.808296074775034, −3.303224610037178, −2.904525343265716, −1.938368379182356, −1.506708345119917, −0.7712776096796075, 0,
0.7712776096796075, 1.506708345119917, 1.938368379182356, 2.904525343265716, 3.303224610037178, 3.808296074775034, 4.596671501232553, 4.948361008288904, 5.562700226737648, 6.179441726875300, 6.428170193989252, 6.803292188678253, 7.507348663314552, 7.957955952283416, 8.678934907699866, 8.990858556629126, 9.640556877432695, 10.04214626197160, 10.28767615520016, 10.84275418354196, 11.65104421951223, 11.76767900397470, 12.14613477130651, 12.90716228302175, 13.15322626513276
