L(s) = 1 | + 0.734·2-s + 3.40·3-s − 1.46·4-s + 5-s + 2.49·6-s − 1.69·7-s − 2.54·8-s + 8.56·9-s + 0.734·10-s − 11-s − 4.96·12-s − 2.13·13-s − 1.24·14-s + 3.40·15-s + 1.05·16-s − 0.398·17-s + 6.29·18-s + 7.29·19-s − 1.46·20-s − 5.74·21-s − 0.734·22-s + 4.28·23-s − 8.64·24-s + 25-s − 1.56·26-s + 18.9·27-s + 2.46·28-s + ⋯ |
L(s) = 1 | + 0.519·2-s + 1.96·3-s − 0.730·4-s + 0.447·5-s + 1.01·6-s − 0.639·7-s − 0.898·8-s + 2.85·9-s + 0.232·10-s − 0.301·11-s − 1.43·12-s − 0.592·13-s − 0.332·14-s + 0.877·15-s + 0.263·16-s − 0.0965·17-s + 1.48·18-s + 1.67·19-s − 0.326·20-s − 1.25·21-s − 0.156·22-s + 0.893·23-s − 1.76·24-s + 0.200·25-s − 0.307·26-s + 3.63·27-s + 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.504852025\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.504852025\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 0.734T + 2T^{2} \) |
| 3 | \( 1 - 3.40T + 3T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 0.398T + 17T^{2} \) |
| 19 | \( 1 - 7.29T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 - 0.172T + 29T^{2} \) |
| 31 | \( 1 + 5.71T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.836T + 41T^{2} \) |
| 43 | \( 1 - 9.14T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 + 5.15T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 + 0.842T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−8.628185640844306771089156338194, −7.64173887852094938919444813005, −7.34045018743967598818632006106, −6.22774383956303873326901752186, −5.23138694552791864895306367334, −4.49862981158398757949659193004, −3.64501125062959110468198684885, −3.01011719993370853571878952522, −2.45798440506325324229509851844, −1.09083899357102492523783566879,
1.09083899357102492523783566879, 2.45798440506325324229509851844, 3.01011719993370853571878952522, 3.64501125062959110468198684885, 4.49862981158398757949659193004, 5.23138694552791864895306367334, 6.22774383956303873326901752186, 7.34045018743967598818632006106, 7.64173887852094938919444813005, 8.628185640844306771089156338194