| L(s) = 1 | − 3.20·3-s − 4.54·7-s + 7.29·9-s + 4.59·11-s + 5.17·13-s + 3.33·17-s − 4.43·19-s + 14.5·21-s + 23-s − 13.8·27-s + 4.13·29-s + 7.82·31-s − 14.7·33-s − 1.03·37-s − 16.6·39-s − 5.75·41-s + 1.67·43-s + 6.20·47-s + 13.6·49-s − 10.7·51-s − 7.35·53-s + 14.2·57-s + 1.83·59-s + 0.524·61-s − 33.1·63-s − 2.55·67-s − 3.20·69-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 1.71·7-s + 2.43·9-s + 1.38·11-s + 1.43·13-s + 0.809·17-s − 1.01·19-s + 3.18·21-s + 0.208·23-s − 2.65·27-s + 0.766·29-s + 1.40·31-s − 2.56·33-s − 0.169·37-s − 2.66·39-s − 0.899·41-s + 0.255·43-s + 0.904·47-s + 1.95·49-s − 1.50·51-s − 1.00·53-s + 1.88·57-s + 0.239·59-s + 0.0671·61-s − 4.18·63-s − 0.312·67-s − 0.386·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8732155027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8732155027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 7 | \( 1 + 4.54T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 7.35T + 53T^{2} \) |
| 59 | \( 1 - 1.83T + 59T^{2} \) |
| 61 | \( 1 - 0.524T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 2.41T + 73T^{2} \) |
| 79 | \( 1 + 9.85T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 5.13T + 89T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−8.416094578261003332584535626727, −7.07795087026910177865457609475, −6.63400208569382515480483760207, −6.09397561488029189340610081465, −5.82021740383290124275276931028, −4.56794058351081821709355875114, −3.95085420612551419694515831169, −3.14775332870361375037803030200, −1.42708999805378117871070638944, −0.63256133730369485597925923204,
0.63256133730369485597925923204, 1.42708999805378117871070638944, 3.14775332870361375037803030200, 3.95085420612551419694515831169, 4.56794058351081821709355875114, 5.82021740383290124275276931028, 6.09397561488029189340610081465, 6.63400208569382515480483760207, 7.07795087026910177865457609475, 8.416094578261003332584535626727
