| L(s) = 1 | − 5-s − 6·11-s + 13-s − 2·17-s − 2·19-s + 2·23-s + 25-s − 10·29-s + 6·31-s + 6·37-s − 2·41-s + 10·43-s − 14·53-s + 6·55-s − 2·59-s − 2·61-s − 65-s + 8·67-s + 10·71-s − 2·73-s + 4·79-s − 4·83-s + 2·85-s − 2·89-s + 2·95-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.80·11-s + 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.986·37-s − 0.312·41-s + 1.52·43-s − 1.92·53-s + 0.809·55-s − 0.260·59-s − 0.256·61-s − 0.124·65-s + 0.977·67-s + 1.18·71-s − 0.234·73-s + 0.450·79-s − 0.439·83-s + 0.216·85-s − 0.211·89-s + 0.205·95-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.01332449598237, −12.76765645534518, −12.45065981124848, −11.64194218263865, −11.20604545622991, −10.86632410431658, −10.56220294788649, −9.914532475873858, −9.326371773069019, −9.093953448497357, −8.172347379715470, −8.003446983319758, −7.719621982624651, −6.930560809949510, −6.615365257533771, −5.822258703748175, −5.517724568401044, −4.920499683906113, −4.391227221377917, −3.949932785924468, −3.198113214716254, −2.711723017825653, −2.251992729069150, −1.508600863583103, −0.6164927312815640, 0,
0.6164927312815640, 1.508600863583103, 2.251992729069150, 2.711723017825653, 3.198113214716254, 3.949932785924468, 4.391227221377917, 4.920499683906113, 5.517724568401044, 5.822258703748175, 6.615365257533771, 6.930560809949510, 7.719621982624651, 8.003446983319758, 8.172347379715470, 9.093953448497357, 9.326371773069019, 9.914532475873858, 10.56220294788649, 10.86632410431658, 11.20604545622991, 11.64194218263865, 12.45065981124848, 12.76765645534518, 13.01332449598237
