| L(s) = 1 | + 3-s + 1.55·5-s + 2.96·7-s + 9-s − 2.24·11-s + 1.55·15-s − 1.01·17-s − 5.35·19-s + 2.96·21-s + 0.782·23-s − 2.58·25-s + 27-s + 2.69·29-s + 7.28·31-s − 2.24·33-s + 4.60·35-s + 7.80·37-s + 5.34·41-s + 3.12·43-s + 1.55·45-s + 7.13·47-s + 1.76·49-s − 1.01·51-s + 13.8·53-s − 3.49·55-s − 5.35·57-s + 4.35·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.695·5-s + 1.11·7-s + 0.333·9-s − 0.677·11-s + 0.401·15-s − 0.247·17-s − 1.22·19-s + 0.646·21-s + 0.163·23-s − 0.516·25-s + 0.192·27-s + 0.499·29-s + 1.30·31-s − 0.391·33-s + 0.777·35-s + 1.28·37-s + 0.834·41-s + 0.476·43-s + 0.231·45-s + 1.04·47-s + 0.252·49-s − 0.142·51-s + 1.90·53-s − 0.471·55-s − 0.709·57-s + 0.566·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.093544812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.093544812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 0.782T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 7.80T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.30T + 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 - 7.67T + 83T^{2} \) |
| 89 | \( 1 + 0.494T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.275785966037408430440684645923, −7.981944532692403253616331561851, −7.04901901200349857427531176626, −6.19199717585738068254980305191, −5.45022091752858175348651500251, −4.59134079638119837803178159086, −4.00087760958921151533616211860, −2.53648176106656397401871714472, −2.24988640614528893375488170324, −1.01916964023970483166641290084,
1.01916964023970483166641290084, 2.24988640614528893375488170324, 2.53648176106656397401871714472, 4.00087760958921151533616211860, 4.59134079638119837803178159086, 5.45022091752858175348651500251, 6.19199717585738068254980305191, 7.04901901200349857427531176626, 7.981944532692403253616331561851, 8.275785966037408430440684645923
