| L(s) = 1 | − 5.65·3-s + 23.0·9-s + 16.9·11-s − 2·17-s + 16.9·19-s + 25·25-s − 79.1·27-s − 96·33-s − 46·41-s + 84.8·43-s + 49·49-s + 11.3·51-s − 96·57-s + 84.8·59-s − 118.·67-s + 142·73-s − 141.·75-s + 241.·81-s − 50.9·83-s − 146·89-s + 94·97-s + 390.·99-s − 118.·107-s + 98·113-s + ⋯ |
| L(s) = 1 | − 1.88·3-s + 2.55·9-s + 1.54·11-s − 0.117·17-s + 0.893·19-s + 25-s − 2.93·27-s − 2.90·33-s − 1.12·41-s + 1.97·43-s + 0.999·49-s + 0.221·51-s − 1.68·57-s + 1.43·59-s − 1.77·67-s + 1.94·73-s − 1.88·75-s + 2.97·81-s − 0.613·83-s − 1.64·89-s + 0.969·97-s + 3.94·99-s − 1.11·107-s + 0.867·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8396492127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8396492127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 5.65T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 2T + 289T^{2} \) |
| 19 | \( 1 - 16.9T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 46T + 1.68e3T^{2} \) |
| 43 | \( 1 - 84.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 142T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 146T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−12.69602036052100301427464019525, −11.91844591660938789248986235130, −11.25806410973001562216721794998, −10.25182571858187948810604443315, −9.139638341863787751974343278040, −7.21773965175082882891014266140, −6.38976711614680935415344241130, −5.32229107117865443618969485744, −4.10067531972306356336740149514, −1.08021558842548084218516329390,
1.08021558842548084218516329390, 4.10067531972306356336740149514, 5.32229107117865443618969485744, 6.38976711614680935415344241130, 7.21773965175082882891014266140, 9.139638341863787751974343278040, 10.25182571858187948810604443315, 11.25806410973001562216721794998, 11.91844591660938789248986235130, 12.69602036052100301427464019525
