L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 3·11-s + 2·12-s + 4·13-s + 14-s + 16-s − 3·17-s + 18-s + 2·19-s + 2·21-s + 3·22-s − 6·23-s + 2·24-s + 4·26-s − 4·27-s + 28-s + 3·29-s + 5·31-s + 32-s + 6·33-s − 3·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s + 0.436·21-s + 0.639·22-s − 1.25·23-s + 0.408·24-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s + 1.04·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.432411577\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.432411577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−9.179198904473865232269004873691, −8.239017656365015125223849506060, −7.985698124817949818204634866172, −6.67937370762302807973390678508, −6.19187304778673505127010809648, −5.01445637294381632948956822464, −4.03128883772338599220444916109, −3.47532993011179219314194223001, −2.43092253567530754938103868933, −1.45408956239590776662905952332,
1.45408956239590776662905952332, 2.43092253567530754938103868933, 3.47532993011179219314194223001, 4.03128883772338599220444916109, 5.01445637294381632948956822464, 6.19187304778673505127010809648, 6.67937370762302807973390678508, 7.985698124817949818204634866172, 8.239017656365015125223849506060, 9.179198904473865232269004873691