| L(s) = 1 | − 2·2-s + 3-s + 4·4-s + 5·5-s − 2·6-s − 8·8-s − 26·9-s − 10·10-s − 2·11-s + 4·12-s + 8·13-s + 5·15-s + 16·16-s + 52·17-s + 52·18-s − 26·19-s + 20·20-s + 4·22-s + 67·23-s − 8·24-s + 25·25-s − 16·26-s − 53·27-s + 69·29-s − 10·30-s + 332·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s − 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.0548·11-s + 0.0962·12-s + 0.170·13-s + 0.0860·15-s + 1/4·16-s + 0.741·17-s + 0.680·18-s − 0.313·19-s + 0.223·20-s + 0.0387·22-s + 0.607·23-s − 0.0680·24-s + 1/5·25-s − 0.120·26-s − 0.377·27-s + 0.441·29-s − 0.0608·30-s + 1.92·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.490156775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490156775\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 52 T + p^{3} T^{2} \) |
| 19 | \( 1 + 26 T + p^{3} T^{2} \) |
| 23 | \( 1 - 67 T + p^{3} T^{2} \) |
| 29 | \( 1 - 69 T + p^{3} T^{2} \) |
| 31 | \( 1 - 332 T + p^{3} T^{2} \) |
| 37 | \( 1 - 196 T + p^{3} T^{2} \) |
| 41 | \( 1 + 353 T + p^{3} T^{2} \) |
| 43 | \( 1 + 369 T + p^{3} T^{2} \) |
| 47 | \( 1 + 88 T + p^{3} T^{2} \) |
| 53 | \( 1 - 582 T + p^{3} T^{2} \) |
| 59 | \( 1 - 350 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 - 291 T + p^{3} T^{2} \) |
| 71 | \( 1 - 770 T + p^{3} T^{2} \) |
| 73 | \( 1 + 628 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 83 | \( 1 + 525 T + p^{3} T^{2} \) |
| 89 | \( 1 + p T + p^{3} T^{2} \) |
| 97 | \( 1 - 290 T + p^{3} T^{2} \) |
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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
−10.31326811407164678881123813400, −9.728138980797757092857073038461, −8.600425776644537930212052514150, −8.195547144602069151204134805798, −6.90660916463151409284539033435, −6.04306578245275794700473369377, −5.00759978637281288340496691006, −3.34471378606230485181948214199, −2.32829999462431807938763059265, −0.841771476100948610717919718941,
0.841771476100948610717919718941, 2.32829999462431807938763059265, 3.34471378606230485181948214199, 5.00759978637281288340496691006, 6.04306578245275794700473369377, 6.90660916463151409284539033435, 8.195547144602069151204134805798, 8.600425776644537930212052514150, 9.728138980797757092857073038461, 10.31326811407164678881123813400
