| L(s) = 1 | − 16·7-s − 36·11-s + 42·13-s − 110·17-s − 116·19-s + 16·23-s − 198·29-s + 240·31-s + 258·37-s − 442·41-s + 292·43-s + 392·47-s − 87·49-s + 142·53-s + 348·59-s − 570·61-s − 692·67-s − 168·71-s + 134·73-s + 576·77-s + 784·79-s + 564·83-s − 1.03e3·89-s − 672·91-s + 382·97-s + 674·101-s + 992·103-s + ⋯ |
| L(s) = 1 | − 0.863·7-s − 0.986·11-s + 0.896·13-s − 1.56·17-s − 1.40·19-s + 0.145·23-s − 1.26·29-s + 1.39·31-s + 1.14·37-s − 1.68·41-s + 1.03·43-s + 1.21·47-s − 0.253·49-s + 0.368·53-s + 0.767·59-s − 1.19·61-s − 1.26·67-s − 0.280·71-s + 0.214·73-s + 0.852·77-s + 1.11·79-s + 0.745·83-s − 1.23·89-s − 0.774·91-s + 0.399·97-s + 0.664·101-s + 0.948·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.061907610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061907610\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 110 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 240 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + 442 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 692 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 134 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−8.851753834284614091392857368247, −8.276090193346409480076597597649, −7.27047022555279141619102725140, −6.40697515062134833477875001115, −5.92580119689374399210033020739, −4.71396372682087169102626217987, −3.95307759304663441299461447116, −2.87091582790039404897616209390, −2.02558001185043872110530288433, −0.45968537237450876577772116559,
0.45968537237450876577772116559, 2.02558001185043872110530288433, 2.87091582790039404897616209390, 3.95307759304663441299461447116, 4.71396372682087169102626217987, 5.92580119689374399210033020739, 6.40697515062134833477875001115, 7.27047022555279141619102725140, 8.276090193346409480076597597649, 8.851753834284614091392857368247
