L(s) = 1 | − 2-s + 3.37·3-s + 4-s − 4.17·5-s − 3.37·6-s + 3.60·7-s − 8-s + 8.41·9-s + 4.17·10-s + 3.37·12-s + 0.669·13-s − 3.60·14-s − 14.0·15-s + 16-s − 4.10·17-s − 8.41·18-s + 19-s − 4.17·20-s + 12.1·21-s + 6.35·23-s − 3.37·24-s + 12.3·25-s − 0.669·26-s + 18.2·27-s + 3.60·28-s − 3.27·29-s + 14.0·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.95·3-s + 0.5·4-s − 1.86·5-s − 1.37·6-s + 1.36·7-s − 0.353·8-s + 2.80·9-s + 1.31·10-s + 0.975·12-s + 0.185·13-s − 0.964·14-s − 3.63·15-s + 0.250·16-s − 0.995·17-s − 1.98·18-s + 0.229·19-s − 0.932·20-s + 2.65·21-s + 1.32·23-s − 0.689·24-s + 2.47·25-s − 0.131·26-s + 3.52·27-s + 0.681·28-s − 0.607·29-s + 2.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.640333027\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.640333027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 3.37T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 - 0.669T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 - 0.862T + 47T^{2} \) |
| 53 | \( 1 - 6.79T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 + 0.503T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 - 0.809T + 97T^{2} \) |
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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
−8.326865365463238705480246703230, −7.77417064098154571348702638020, −7.36598034124865073538993272529, −6.79822940453174320154493093599, −4.95580769673339845963982573754, −4.32200081137629643596369316143, −3.64909916777546020898410194665, −2.88708825428656759975366117240, −1.96527698299693861899827677529, −0.963223405179308908906659801897,
0.963223405179308908906659801897, 1.96527698299693861899827677529, 2.88708825428656759975366117240, 3.64909916777546020898410194665, 4.32200081137629643596369316143, 4.95580769673339845963982573754, 6.79822940453174320154493093599, 7.36598034124865073538993272529, 7.77417064098154571348702638020, 8.326865365463238705480246703230