| L(s) = 1 | − 3-s − 1.91·5-s − 2.53·7-s + 9-s − 2.79·11-s + 2.05·13-s + 1.91·15-s + 6.58·17-s + 0.477·19-s + 2.53·21-s − 23-s − 1.32·25-s − 27-s − 29-s + 2.03·31-s + 2.79·33-s + 4.86·35-s − 11.2·37-s − 2.05·39-s − 4.45·41-s + 6.32·43-s − 1.91·45-s − 5.49·47-s − 0.576·49-s − 6.58·51-s + 0.163·53-s + 5.36·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.857·5-s − 0.957·7-s + 0.333·9-s − 0.843·11-s + 0.570·13-s + 0.495·15-s + 1.59·17-s + 0.109·19-s + 0.553·21-s − 0.208·23-s − 0.264·25-s − 0.192·27-s − 0.185·29-s + 0.364·31-s + 0.486·33-s + 0.821·35-s − 1.85·37-s − 0.329·39-s − 0.695·41-s + 0.964·43-s − 0.285·45-s − 0.802·47-s − 0.0823·49-s − 0.922·51-s + 0.0224·53-s + 0.723·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6739174328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6739174328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 - 0.477T + 19T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 - 0.163T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 3.48T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 - 5.70T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.77964404454101228244712455974, −7.17475628128665097746712173875, −6.45042702355195224429628665151, −5.68944735904822521429683559290, −5.20934786962596368087375780947, −4.19860635305037199783321786463, −3.48176137855063921411377544593, −2.96679215139818032832627240531, −1.59275708354062512206419150606, −0.42216584412526416989265312104,
0.42216584412526416989265312104, 1.59275708354062512206419150606, 2.96679215139818032832627240531, 3.48176137855063921411377544593, 4.19860635305037199783321786463, 5.20934786962596368087375780947, 5.68944735904822521429683559290, 6.45042702355195224429628665151, 7.17475628128665097746712173875, 7.77964404454101228244712455974
