| L(s) = 1 | − 0.667·2-s − 3.21·3-s − 1.55·4-s − 5-s + 2.14·6-s + 2.82·7-s + 2.37·8-s + 7.31·9-s + 0.667·10-s + 3.25·11-s + 4.99·12-s − 2.24·13-s − 1.88·14-s + 3.21·15-s + 1.52·16-s + 3.79·17-s − 4.88·18-s + 5.30·19-s + 1.55·20-s − 9.08·21-s − 2.17·22-s + 0.878·23-s − 7.62·24-s + 25-s + 1.49·26-s − 13.8·27-s − 4.39·28-s + ⋯ |
| L(s) = 1 | − 0.471·2-s − 1.85·3-s − 0.777·4-s − 0.447·5-s + 0.875·6-s + 1.06·7-s + 0.838·8-s + 2.43·9-s + 0.211·10-s + 0.981·11-s + 1.44·12-s − 0.622·13-s − 0.504·14-s + 0.829·15-s + 0.381·16-s + 0.919·17-s − 1.15·18-s + 1.21·19-s + 0.347·20-s − 1.98·21-s − 0.463·22-s + 0.183·23-s − 1.55·24-s + 0.200·25-s + 0.293·26-s − 2.67·27-s − 0.831·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6764106453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6764106453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.667T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 - 0.878T + 23T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 0.286T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 - 9.09T + 73T^{2} \) |
| 79 | \( 1 - 1.95T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 9.29T + 97T^{2} \) |
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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
−9.429958921439797758806025059502, −8.247817941985690326354899978647, −7.55777766223232297733500529561, −6.89227747690907120534021937226, −5.79636612723551602605998608004, −5.03652133254682313213038669099, −4.63667040948873718906999285912, −3.68240029110603654440662264828, −1.45229341449627366037711914893, −0.74893579062944820788614049697,
0.74893579062944820788614049697, 1.45229341449627366037711914893, 3.68240029110603654440662264828, 4.63667040948873718906999285912, 5.03652133254682313213038669099, 5.79636612723551602605998608004, 6.89227747690907120534021937226, 7.55777766223232297733500529561, 8.247817941985690326354899978647, 9.429958921439797758806025059502
