| L(s) = 1 | − 3-s − 5-s − 2.82·7-s + 9-s + 2.82·11-s + 0.828·13-s + 15-s − 7.65·17-s − 19-s + 2.82·21-s + 4·23-s + 25-s − 27-s − 0.828·29-s − 2.82·33-s + 2.82·35-s − 4.82·37-s − 0.828·39-s − 0.828·41-s + 2.82·43-s − 45-s + 4·47-s + 1.00·49-s + 7.65·51-s − 2·53-s − 2.82·55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.06·7-s + 0.333·9-s + 0.852·11-s + 0.229·13-s + 0.258·15-s − 1.85·17-s − 0.229·19-s + 0.617·21-s + 0.834·23-s + 0.200·25-s − 0.192·27-s − 0.153·29-s − 0.492·33-s + 0.478·35-s − 0.793·37-s − 0.132·39-s − 0.129·41-s + 0.431·43-s − 0.149·45-s + 0.583·47-s + 0.142·49-s + 1.07·51-s − 0.274·53-s − 0.381·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9325732298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9325732298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 + 4.82T + 97T^{2} \) |
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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
−9.064853322519666626477245508193, −8.402341790342959176459082174420, −7.12537956973785423526028238406, −6.71777962210647024934771792138, −6.08858926564127662291794786963, −4.99348905177960789715348605884, −4.12897135163360624663474684641, −3.40523892506397326240237129128, −2.15066712598884637300419173657, −0.62731283227951110386030647188,
0.62731283227951110386030647188, 2.15066712598884637300419173657, 3.40523892506397326240237129128, 4.12897135163360624663474684641, 4.99348905177960789715348605884, 6.08858926564127662291794786963, 6.71777962210647024934771792138, 7.12537956973785423526028238406, 8.402341790342959176459082174420, 9.064853322519666626477245508193
