| L(s) = 1 | + 3·3-s + 10·7-s + 9·9-s − 46·11-s − 34·13-s + 66·17-s + 104·19-s + 30·21-s + 164·23-s + 27·27-s + 224·29-s − 72·31-s − 138·33-s − 22·37-s − 102·39-s + 194·41-s + 108·43-s − 480·47-s − 243·49-s + 198·51-s + 286·53-s + 312·57-s + 426·59-s + 698·61-s + 90·63-s + 328·67-s + 492·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.539·7-s + 1/3·9-s − 1.26·11-s − 0.725·13-s + 0.941·17-s + 1.25·19-s + 0.311·21-s + 1.48·23-s + 0.192·27-s + 1.43·29-s − 0.417·31-s − 0.727·33-s − 0.0977·37-s − 0.418·39-s + 0.738·41-s + 0.383·43-s − 1.48·47-s − 0.708·49-s + 0.543·51-s + 0.741·53-s + 0.725·57-s + 0.940·59-s + 1.46·61-s + 0.179·63-s + 0.598·67-s + 0.858·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.593442329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.593442329\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 46 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 164 T + p^{3} T^{2} \) |
| 29 | \( 1 - 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 22 T + p^{3} T^{2} \) |
| 41 | \( 1 - 194 T + p^{3} T^{2} \) |
| 43 | \( 1 - 108 T + p^{3} T^{2} \) |
| 47 | \( 1 + 480 T + p^{3} T^{2} \) |
| 53 | \( 1 - 286 T + p^{3} T^{2} \) |
| 59 | \( 1 - 426 T + p^{3} T^{2} \) |
| 61 | \( 1 - 698 T + p^{3} T^{2} \) |
| 67 | \( 1 - 328 T + p^{3} T^{2} \) |
| 71 | \( 1 - 188 T + p^{3} T^{2} \) |
| 73 | \( 1 + 740 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 - 412 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1206 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1384 T + p^{3} T^{2} \) |
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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
−10.14383656026437583208601465896, −9.472587396040658721478135426770, −8.343569793430208824416798880352, −7.72279826068598687963605007706, −6.94246387812257319000376097238, −5.38338645083082114753325591963, −4.84984809986018237861115917283, −3.32479692060009718445180700753, −2.47238169341871143462295543762, −0.970048750845731928761645008312,
0.970048750845731928761645008312, 2.47238169341871143462295543762, 3.32479692060009718445180700753, 4.84984809986018237861115917283, 5.38338645083082114753325591963, 6.94246387812257319000376097238, 7.72279826068598687963605007706, 8.343569793430208824416798880352, 9.472587396040658721478135426770, 10.14383656026437583208601465896
