L(s) = 1 | − 2.95·3-s − 5-s − 3.95·7-s + 5.74·9-s − 0.957·11-s + 2.74·13-s + 2.95·15-s + 5.74·17-s − 6.74·19-s + 11.7·21-s − 23-s + 25-s − 8.12·27-s − 5.21·29-s + 5.95·31-s + 2.83·33-s + 3.95·35-s − 9.12·37-s − 8.12·39-s + 0.252·41-s + 8·43-s − 5.74·45-s − 5.49·47-s + 8.66·49-s − 16.9·51-s + 7.12·53-s + 0.957·55-s + ⋯ |
L(s) = 1 | − 1.70·3-s − 0.447·5-s − 1.49·7-s + 1.91·9-s − 0.288·11-s + 0.761·13-s + 0.763·15-s + 1.39·17-s − 1.54·19-s + 2.55·21-s − 0.208·23-s + 0.200·25-s − 1.56·27-s − 0.967·29-s + 1.07·31-s + 0.493·33-s + 0.668·35-s − 1.50·37-s − 1.30·39-s + 0.0394·41-s + 1.21·43-s − 0.856·45-s − 0.801·47-s + 1.23·49-s − 2.38·51-s + 0.978·53-s + 0.129·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3402380448\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3402380448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2.95T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 + 0.957T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 9.12T + 37T^{2} \) |
| 41 | \( 1 - 0.252T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 4.78T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.704T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.75628808522258526219474949497, −6.81958857123440871293661366130, −6.54071180711964683928202502563, −5.77513251299169465228876921778, −5.40092164812512944042898637357, −4.29633131407904759341016497101, −3.77048350142978692165784547517, −2.85005657271218686986235037289, −1.41722309508015648566065006397, −0.33867661168378627352240419280,
0.33867661168378627352240419280, 1.41722309508015648566065006397, 2.85005657271218686986235037289, 3.77048350142978692165784547517, 4.29633131407904759341016497101, 5.40092164812512944042898637357, 5.77513251299169465228876921778, 6.54071180711964683928202502563, 6.81958857123440871293661366130, 7.75628808522258526219474949497