| L(s) = 1 | + 2-s − 1.47·3-s + 4-s − 2.50·5-s − 1.47·6-s − 2.19·7-s + 8-s − 0.813·9-s − 2.50·10-s + 11-s − 1.47·12-s − 0.122·13-s − 2.19·14-s + 3.70·15-s + 16-s − 5.80·17-s − 0.813·18-s − 2.80·19-s − 2.50·20-s + 3.24·21-s + 22-s − 2.34·23-s − 1.47·24-s + 1.29·25-s − 0.122·26-s + 5.63·27-s − 2.19·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.853·3-s + 0.5·4-s − 1.12·5-s − 0.603·6-s − 0.828·7-s + 0.353·8-s − 0.271·9-s − 0.793·10-s + 0.301·11-s − 0.426·12-s − 0.0340·13-s − 0.585·14-s + 0.957·15-s + 0.250·16-s − 1.40·17-s − 0.191·18-s − 0.643·19-s − 0.560·20-s + 0.707·21-s + 0.213·22-s − 0.489·23-s − 0.301·24-s + 0.258·25-s − 0.0240·26-s + 1.08·27-s − 0.414·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7099666384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7099666384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 7 | \( 1 + 2.19T + 7T^{2} \) |
| 13 | \( 1 + 0.122T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 0.143T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.388T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 0.454T + 53T^{2} \) |
| 59 | \( 1 + 6.87T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 6.02T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 - 6.47T + 89T^{2} \) |
| 97 | \( 1 - 2.74T + 97T^{2} \) |
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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
−8.231068437930917051930986338028, −7.46538896190952772182260542312, −6.66491404230677272600654126855, −6.20141481678926477183388231309, −5.48267113446187043163285087702, −4.45736257972955835375501941206, −4.02440689706281499127671291107, −3.15022487025806961801481842082, −2.12640381635805103152266788836, −0.41801085742393118341295670618,
0.41801085742393118341295670618, 2.12640381635805103152266788836, 3.15022487025806961801481842082, 4.02440689706281499127671291107, 4.45736257972955835375501941206, 5.48267113446187043163285087702, 6.20141481678926477183388231309, 6.66491404230677272600654126855, 7.46538896190952772182260542312, 8.231068437930917051930986338028
