| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 2·7-s + 3·8-s + 9-s + 12-s − 4·13-s − 2·14-s − 16-s − 2·17-s − 18-s − 2·21-s − 2·23-s − 3·24-s + 4·26-s − 27-s − 2·28-s + 29-s + 4·31-s − 5·32-s + 2·34-s − 36-s − 2·37-s + 4·39-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.436·21-s − 0.417·23-s − 0.612·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 0.185·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 1/6·36-s − 0.328·37-s + 0.640·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.693673905877768281354694093613, −7.938558467480015624601923874544, −7.36032983033591685467342280674, −6.42968104843145663951320739990, −5.31539983467840359135097646602, −4.74266370839170039373555185594, −4.01594438426739229276737033711, −2.43360183743919969020737882343, −1.28767289236790541838025522846, 0,
1.28767289236790541838025522846, 2.43360183743919969020737882343, 4.01594438426739229276737033711, 4.74266370839170039373555185594, 5.31539983467840359135097646602, 6.42968104843145663951320739990, 7.36032983033591685467342280674, 7.938558467480015624601923874544, 8.693673905877768281354694093613
