| L(s) = 1 | + 3-s − 5-s + 4.47·7-s + 9-s + 3.66·11-s + 3.81·13-s − 15-s − 6.96·17-s − 0.0738·19-s + 4.47·21-s − 7.13·23-s + 25-s + 27-s + 0.538·29-s + 2.70·31-s + 3.66·33-s − 4.47·35-s + 8.36·37-s + 3.81·39-s − 0.701·41-s + 10.9·43-s − 45-s + 2.26·47-s + 13.0·49-s − 6.96·51-s + 1.74·53-s − 3.66·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.69·7-s + 0.333·9-s + 1.10·11-s + 1.05·13-s − 0.258·15-s − 1.69·17-s − 0.0169·19-s + 0.976·21-s − 1.48·23-s + 0.200·25-s + 0.192·27-s + 0.100·29-s + 0.485·31-s + 0.637·33-s − 0.756·35-s + 1.37·37-s + 0.610·39-s − 0.109·41-s + 1.66·43-s − 0.149·45-s + 0.330·47-s + 1.86·49-s − 0.975·51-s + 0.240·53-s − 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.375075234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.375075234\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 19 | \( 1 + 0.0738T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 0.538T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 - 8.36T + 37T^{2} \) |
| 41 | \( 1 + 0.701T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 - 1.74T + 53T^{2} \) |
| 59 | \( 1 - 0.807T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 - 3.56T + 83T^{2} \) |
| 89 | \( 1 - 8.57T + 89T^{2} \) |
| 97 | \( 1 - 9.19T + 97T^{2} \) |
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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
−7.944084233399714378382231848734, −7.33900940384737994004210813888, −6.43291244478583211228671105688, −5.88821093828527551079755032990, −4.68818440818732259680056250993, −4.24581654069651171054702663388, −3.79076820916820763183438534601, −2.49355200325312525564979424923, −1.80310609603500606193762052574, −0.945971938827641472057716976106,
0.945971938827641472057716976106, 1.80310609603500606193762052574, 2.49355200325312525564979424923, 3.79076820916820763183438534601, 4.24581654069651171054702663388, 4.68818440818732259680056250993, 5.88821093828527551079755032990, 6.43291244478583211228671105688, 7.33900940384737994004210813888, 7.944084233399714378382231848734
