L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 4·11-s − 2·12-s − 2·13-s + 16-s − 8·17-s − 18-s − 6·19-s + 4·22-s + 4·23-s + 2·24-s + 2·26-s + 4·27-s − 6·29-s + 4·31-s − 32-s + 8·33-s + 8·34-s + 36-s + 10·37-s + 6·38-s + 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s + 0.834·23-s + 0.408·24-s + 0.392·26-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.39·33-s + 1.37·34-s + 1/6·36-s + 1.64·37-s + 0.973·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2747366797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2747366797\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.980008451296710170626375632277, −8.169311792616696532350927810659, −7.42579423059284725914097555874, −6.46853840054551386721568674637, −6.11996720874293404664984988869, −4.97043025034748725774602844716, −4.49802827242680952124977515011, −2.86783321468236434903187431571, −2.02264162411673129083359187598, −0.36857496855728403516977276168,
0.36857496855728403516977276168, 2.02264162411673129083359187598, 2.86783321468236434903187431571, 4.49802827242680952124977515011, 4.97043025034748725774602844716, 6.11996720874293404664984988869, 6.46853840054551386721568674637, 7.42579423059284725914097555874, 8.169311792616696532350927810659, 8.980008451296710170626375632277