| L(s) = 1 | + 5·2-s + 17·4-s − 3·7-s + 45·8-s + 11·11-s − 32·13-s − 15·14-s + 89·16-s + 33·17-s + 47·19-s + 55·22-s + 113·23-s − 160·26-s − 51·28-s + 54·29-s + 178·31-s + 85·32-s + 165·34-s − 19·37-s + 235·38-s − 139·41-s + 308·43-s + 187·44-s + 565·46-s + 195·47-s − 334·49-s − 544·52-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s − 0.161·7-s + 1.98·8-s + 0.301·11-s − 0.682·13-s − 0.286·14-s + 1.39·16-s + 0.470·17-s + 0.567·19-s + 0.533·22-s + 1.02·23-s − 1.20·26-s − 0.344·28-s + 0.345·29-s + 1.03·31-s + 0.469·32-s + 0.832·34-s − 0.0844·37-s + 1.00·38-s − 0.529·41-s + 1.09·43-s + 0.640·44-s + 1.81·46-s + 0.605·47-s − 0.973·49-s − 1.45·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.739644746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.739644746\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 33 T + p^{3} T^{2} \) |
| 19 | \( 1 - 47 T + p^{3} T^{2} \) |
| 23 | \( 1 - 113 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 178 T + p^{3} T^{2} \) |
| 37 | \( 1 + 19 T + p^{3} T^{2} \) |
| 41 | \( 1 + 139 T + p^{3} T^{2} \) |
| 43 | \( 1 - 308 T + p^{3} T^{2} \) |
| 47 | \( 1 - 195 T + p^{3} T^{2} \) |
| 53 | \( 1 - 152 T + p^{3} T^{2} \) |
| 59 | \( 1 - 625 T + p^{3} T^{2} \) |
| 61 | \( 1 - 320 T + p^{3} T^{2} \) |
| 67 | \( 1 + 200 T + p^{3} T^{2} \) |
| 71 | \( 1 - 947 T + p^{3} T^{2} \) |
| 73 | \( 1 - 448 T + p^{3} T^{2} \) |
| 79 | \( 1 + 721 T + p^{3} T^{2} \) |
| 83 | \( 1 - 142 T + p^{3} T^{2} \) |
| 89 | \( 1 + 404 T + p^{3} T^{2} \) |
| 97 | \( 1 + 79 T + p^{3} T^{2} \) |
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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
−8.421877292334333203785891029358, −7.39078927829645175274875409893, −6.85909661335438011320058480485, −6.06552445375834719015444590207, −5.26463006480057123832155644282, −4.70389074008462825490514566466, −3.79284404151435500765286743019, −3.03251736270985456306182652396, −2.26540546028668896082198846582, −0.956056701552747997786135195036,
0.956056701552747997786135195036, 2.26540546028668896082198846582, 3.03251736270985456306182652396, 3.79284404151435500765286743019, 4.70389074008462825490514566466, 5.26463006480057123832155644282, 6.06552445375834719015444590207, 6.85909661335438011320058480485, 7.39078927829645175274875409893, 8.421877292334333203785891029358
