| L(s) = 1 | + 3-s + 0.395·5-s − 3.09·7-s + 9-s − 2.63·11-s − 2.21·13-s + 0.395·15-s + 3.70·17-s + 7.04·19-s − 3.09·21-s − 4.84·25-s + 27-s − 6.67·29-s + 2.31·31-s − 2.63·33-s − 1.22·35-s + 5.71·37-s − 2.21·39-s − 2.81·41-s + 3.91·43-s + 0.395·45-s − 4.37·47-s + 2.56·49-s + 3.70·51-s + 12.6·53-s − 1.04·55-s + 7.04·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.176·5-s − 1.16·7-s + 0.333·9-s − 0.794·11-s − 0.614·13-s + 0.102·15-s + 0.899·17-s + 1.61·19-s − 0.674·21-s − 0.968·25-s + 0.192·27-s − 1.23·29-s + 0.415·31-s − 0.458·33-s − 0.206·35-s + 0.939·37-s − 0.354·39-s − 0.439·41-s + 0.596·43-s + 0.0589·45-s − 0.638·47-s + 0.366·49-s + 0.519·51-s + 1.74·53-s − 0.140·55-s + 0.932·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927903307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927903307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 0.395T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 6.63T + 61T^{2} \) |
| 67 | \( 1 + 1.34T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 + 7.43T + 79T^{2} \) |
| 83 | \( 1 - 0.689T + 83T^{2} \) |
| 89 | \( 1 - 6.16T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−7.80247678925310166711781294140, −7.50062772656555658083174362690, −6.76828978480342727538984950260, −5.70678437600438944334891107739, −5.42992840507247495621257473791, −4.27049142460961416026285784388, −3.38979235991544030374368983377, −2.90807608624196207787846598791, −2.01496636782259636477133426578, −0.68115511272940066502608310526,
0.68115511272940066502608310526, 2.01496636782259636477133426578, 2.90807608624196207787846598791, 3.38979235991544030374368983377, 4.27049142460961416026285784388, 5.42992840507247495621257473791, 5.70678437600438944334891107739, 6.76828978480342727538984950260, 7.50062772656555658083174362690, 7.80247678925310166711781294140
