L(s) = 1 | + 1.05·3-s + 4.18·5-s − 4.01·7-s − 1.89·9-s + 5.45·11-s + 1.75·13-s + 4.40·15-s + 2·17-s + 0.462·19-s − 4.22·21-s + 3.43·23-s + 12.5·25-s − 5.14·27-s + 1.75·29-s − 7.83·31-s + 5.72·33-s − 16.8·35-s − 37-s + 1.84·39-s + 6.46·41-s + 6.40·43-s − 7.94·45-s − 12.8·47-s + 9.15·49-s + 2.10·51-s + 12.0·53-s + 22.8·55-s + ⋯ |
L(s) = 1 | + 0.606·3-s + 1.87·5-s − 1.51·7-s − 0.632·9-s + 1.64·11-s + 0.487·13-s + 1.13·15-s + 0.485·17-s + 0.106·19-s − 0.921·21-s + 0.715·23-s + 2.51·25-s − 0.989·27-s + 0.326·29-s − 1.40·31-s + 0.996·33-s − 2.84·35-s − 0.164·37-s + 0.295·39-s + 1.00·41-s + 0.976·43-s − 1.18·45-s − 1.86·47-s + 1.30·49-s + 0.294·51-s + 1.65·53-s + 3.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.927146658\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927146658\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.462T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 41 | \( 1 - 6.46T + 41T^{2} \) |
| 43 | \( 1 - 6.40T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 7.16T + 59T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 - 1.91T + 83T^{2} \) |
| 89 | \( 1 + 1.44T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.229393525354931374576437661281, −8.635437371336205256162597361616, −7.21669601951434065671727667592, −6.43914144265328392966013618795, −6.04674588469344622964827206956, −5.31020806642198156616599196782, −3.80671296978957882826395424212, −3.13172366494242688707122900147, −2.25838345535973108436648182510, −1.15060345992474559569560401282,
1.15060345992474559569560401282, 2.25838345535973108436648182510, 3.13172366494242688707122900147, 3.80671296978957882826395424212, 5.31020806642198156616599196782, 6.04674588469344622964827206956, 6.43914144265328392966013618795, 7.21669601951434065671727667592, 8.635437371336205256162597361616, 9.229393525354931374576437661281