| L(s) = 1 | − 2·3-s − 5-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 4·19-s + 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s − 2·37-s − 4·39-s − 6·41-s + 10·43-s − 45-s + 6·47-s − 12·51-s + 6·53-s + 8·57-s + 12·59-s + 2·61-s − 2·65-s − 2·67-s − 12·69-s − 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.244·67-s − 1.44·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.215592430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215592430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.13123074367083, −15.56777343889858, −14.77025266716560, −14.62209208444951, −13.63400844647978, −13.13238392843479, −12.46378778587444, −12.01737156040302, −11.56179442794275, −10.87159422377451, −10.59642056439151, −9.939909246597060, −9.060661584083742, −8.581213127948732, −7.847715038972807, −7.176709441408121, −6.636251296686564, −5.842648446183121, −5.498436186071081, −4.793998383172877, −4.017746404932287, −3.362938118646155, −2.471187497605137, −1.260410108030976, −0.5782744470046328,
0.5782744470046328, 1.260410108030976, 2.471187497605137, 3.362938118646155, 4.017746404932287, 4.793998383172877, 5.498436186071081, 5.842648446183121, 6.636251296686564, 7.176709441408121, 7.847715038972807, 8.581213127948732, 9.060661584083742, 9.939909246597060, 10.59642056439151, 10.87159422377451, 11.56179442794275, 12.01737156040302, 12.46378778587444, 13.13238392843479, 13.63400844647978, 14.62209208444951, 14.77025266716560, 15.56777343889858, 16.13123074367083
