| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 13-s + 16-s + 7·17-s + 3·19-s − 22-s − 26-s + 4·29-s − 6·31-s + 32-s + 7·34-s + 8·37-s + 3·38-s − 5·41-s + 4·43-s − 44-s − 12·47-s − 52-s − 10·53-s + 4·58-s + 4·59-s − 8·61-s − 6·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 0.277·13-s + 1/4·16-s + 1.69·17-s + 0.688·19-s − 0.213·22-s − 0.196·26-s + 0.742·29-s − 1.07·31-s + 0.176·32-s + 1.20·34-s + 1.31·37-s + 0.486·38-s − 0.780·41-s + 0.609·43-s − 0.150·44-s − 1.75·47-s − 0.138·52-s − 1.37·53-s + 0.525·58-s + 0.520·59-s − 1.02·61-s − 0.762·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.662553834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.662553834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.70138642204867, −12.33890136078138, −11.96570416445023, −11.27936380522933, −11.13558288481169, −10.46717673942134, −9.932961763000702, −9.610198577148737, −9.232400179841361, −8.306735967541562, −7.922696550927198, −7.740330621184413, −6.975213076415880, −6.639389741680247, −5.988603937548198, −5.549891667237803, −5.062688455831063, −4.766429508065323, −3.964187624896898, −3.498588166288854, −3.045476134082578, −2.554983460211548, −1.780895088903540, −1.234176925171170, −0.5213465168620561,
0.5213465168620561, 1.234176925171170, 1.780895088903540, 2.554983460211548, 3.045476134082578, 3.498588166288854, 3.964187624896898, 4.766429508065323, 5.062688455831063, 5.549891667237803, 5.988603937548198, 6.639389741680247, 6.975213076415880, 7.740330621184413, 7.922696550927198, 8.306735967541562, 9.232400179841361, 9.610198577148737, 9.932961763000702, 10.46717673942134, 11.13558288481169, 11.27936380522933, 11.96570416445023, 12.33890136078138, 12.70138642204867
