| L(s) = 1 | + 3-s + 3.59·5-s + 4.10·7-s + 9-s + 1.62·11-s + 5.55·13-s + 3.59·15-s − 5.51·17-s − 0.688·19-s + 4.10·21-s − 4.57·23-s + 7.89·25-s + 27-s + 3.57·29-s + 7.18·31-s + 1.62·33-s + 14.7·35-s − 10.0·37-s + 5.55·39-s − 9.17·41-s + 2.89·43-s + 3.59·45-s − 5.70·47-s + 9.87·49-s − 5.51·51-s − 0.787·53-s + 5.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.60·5-s + 1.55·7-s + 0.333·9-s + 0.488·11-s + 1.53·13-s + 0.927·15-s − 1.33·17-s − 0.158·19-s + 0.896·21-s − 0.954·23-s + 1.57·25-s + 0.192·27-s + 0.663·29-s + 1.29·31-s + 0.282·33-s + 2.49·35-s − 1.65·37-s + 0.889·39-s − 1.43·41-s + 0.440·43-s + 0.535·45-s − 0.831·47-s + 1.41·49-s − 0.772·51-s − 0.108·53-s + 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.288514617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288514617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 0.688T + 19T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 + 0.787T + 53T^{2} \) |
| 59 | \( 1 + 2.75T + 59T^{2} \) |
| 61 | \( 1 + 1.60T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 + 6.98T + 71T^{2} \) |
| 73 | \( 1 - 0.181T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 6.78T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−8.711763708025683807863053486209, −7.996437650466320641699827481921, −6.74618109323036460733216954075, −6.35202038633570784630257677746, −5.46195348534082044812765020568, −4.69585837944775024139494354215, −3.93704746992703681161764042267, −2.71414287707981973402470959463, −1.74283436535647703138444855752, −1.45674301598096392611951289615,
1.45674301598096392611951289615, 1.74283436535647703138444855752, 2.71414287707981973402470959463, 3.93704746992703681161764042267, 4.69585837944775024139494354215, 5.46195348534082044812765020568, 6.35202038633570784630257677746, 6.74618109323036460733216954075, 7.996437650466320641699827481921, 8.711763708025683807863053486209
