| L(s) = 1 | + 2-s + 0.470·3-s + 4-s + 5-s + 0.470·6-s + 2.26·7-s + 8-s − 2.77·9-s + 10-s + 2.65·11-s + 0.470·12-s − 5.41·13-s + 2.26·14-s + 0.470·15-s + 16-s + 2.57·17-s − 2.77·18-s − 5.28·19-s + 20-s + 1.06·21-s + 2.65·22-s + 6.93·23-s + 0.470·24-s + 25-s − 5.41·26-s − 2.72·27-s + 2.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.271·3-s + 0.5·4-s + 0.447·5-s + 0.192·6-s + 0.855·7-s + 0.353·8-s − 0.926·9-s + 0.316·10-s + 0.800·11-s + 0.135·12-s − 1.50·13-s + 0.604·14-s + 0.121·15-s + 0.250·16-s + 0.623·17-s − 0.654·18-s − 1.21·19-s + 0.223·20-s + 0.232·21-s + 0.565·22-s + 1.44·23-s + 0.0961·24-s + 0.200·25-s − 1.06·26-s − 0.523·27-s + 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.922950929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.922950929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.470T + 3T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 0.582T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 8.88T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 5.33T + 97T^{2} \) |
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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
−8.442298192864913235129803029855, −7.60755540093375474008058355982, −6.98191152846410599704694997720, −6.05302120912685948957080135263, −5.44281831469250542349192988250, −4.66673872688043098637236558094, −4.01084800644896473891394100202, −2.70923915453652366081714977469, −2.38225621247572665543665084701, −1.04964069528047849824379043971,
1.04964069528047849824379043971, 2.38225621247572665543665084701, 2.70923915453652366081714977469, 4.01084800644896473891394100202, 4.66673872688043098637236558094, 5.44281831469250542349192988250, 6.05302120912685948957080135263, 6.98191152846410599704694997720, 7.60755540093375474008058355982, 8.442298192864913235129803029855
