| L(s) = 1 | − 2·7-s + 2·13-s − 3·17-s − 2·19-s − 9·23-s − 5·31-s + 2·37-s − 3·41-s − 43-s + 3·47-s − 3·49-s + 3·53-s − 3·59-s + 2·61-s − 5·67-s + 6·71-s + 8·73-s + 79-s − 4·91-s − 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.554·13-s − 0.727·17-s − 0.458·19-s − 1.87·23-s − 0.898·31-s + 0.328·37-s − 0.468·41-s − 0.152·43-s + 0.437·47-s − 3/7·49-s + 0.412·53-s − 0.390·59-s + 0.256·61-s − 0.610·67-s + 0.712·71-s + 0.936·73-s + 0.112·79-s − 0.419·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6014261978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6014261978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.27585322013508, −12.85246702357800, −12.41505428458716, −11.94281756930419, −11.35761362673247, −10.96575786981979, −10.39864446595711, −9.961863297084352, −9.531682659870629, −8.925733611596292, −8.575872490275043, −7.955388784834296, −7.538631923329937, −6.797507449094359, −6.445924593135990, −5.983420450883396, −5.505956397096723, −4.767226690869726, −4.179287068225792, −3.708453481104860, −3.258926926713391, −2.365997873559661, −2.029496368498127, −1.206031151591480, −0.2295529731698680,
0.2295529731698680, 1.206031151591480, 2.029496368498127, 2.365997873559661, 3.258926926713391, 3.708453481104860, 4.179287068225792, 4.767226690869726, 5.505956397096723, 5.983420450883396, 6.445924593135990, 6.797507449094359, 7.538631923329937, 7.955388784834296, 8.575872490275043, 8.925733611596292, 9.531682659870629, 9.961863297084352, 10.39864446595711, 10.96575786981979, 11.35761362673247, 11.94281756930419, 12.41505428458716, 12.85246702357800, 13.27585322013508
