| L(s) = 1 | − 0.836·3-s − 2.30·9-s + 0.403·11-s − 6.20·13-s − 2.62·17-s − 6.41·19-s + 6.54·23-s + 4.43·27-s + 1.96·29-s − 4.66·31-s − 0.337·33-s − 3.44·37-s + 5.19·39-s − 2.68·41-s − 10.8·43-s + 9.67·47-s + 2.19·51-s − 5.97·53-s + 5.36·57-s − 9.18·59-s − 5.69·61-s + 11.8·67-s − 5.47·69-s − 0.530·71-s − 8.20·73-s − 9.12·79-s + 3.19·81-s + ⋯ |
| L(s) = 1 | − 0.483·3-s − 0.766·9-s + 0.121·11-s − 1.72·13-s − 0.636·17-s − 1.47·19-s + 1.36·23-s + 0.853·27-s + 0.364·29-s − 0.838·31-s − 0.0587·33-s − 0.566·37-s + 0.831·39-s − 0.418·41-s − 1.65·43-s + 1.41·47-s + 0.307·51-s − 0.820·53-s + 0.710·57-s − 1.19·59-s − 0.729·61-s + 1.45·67-s − 0.659·69-s − 0.0629·71-s − 0.959·73-s − 1.02·79-s + 0.354·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5794358457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5794358457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.836T + 3T^{2} \) |
| 11 | \( 1 - 0.403T + 11T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 0.530T + 71T^{2} \) |
| 73 | \( 1 + 8.20T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.54541054307026640118861129557, −6.88995587990563269646098297190, −6.41320022633532190196882777000, −5.54998817742693256257206218387, −4.90277322774387611346427239033, −4.45734160690639086109043526383, −3.30906804178214848591899429604, −2.58973189459503510444151046168, −1.80261654403250365911803326341, −0.35061132203019140837799056427,
0.35061132203019140837799056427, 1.80261654403250365911803326341, 2.58973189459503510444151046168, 3.30906804178214848591899429604, 4.45734160690639086109043526383, 4.90277322774387611346427239033, 5.54998817742693256257206218387, 6.41320022633532190196882777000, 6.88995587990563269646098297190, 7.54541054307026640118861129557
