| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 3·14-s + 16-s − 7·17-s + 18-s + 5·19-s + 3·21-s + 22-s − 23-s + 24-s + 4·26-s + 27-s + 3·28-s − 10·29-s + 2·31-s + 32-s + 33-s − 7·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 1.14·19-s + 0.654·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.566·28-s − 1.85·29-s + 0.359·31-s + 0.176·32-s + 0.174·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.878162914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.878162914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.076343222689528847062851881906, −8.677792325242591825827135877996, −7.67832782283085308061068242045, −7.05704044075678616417823809938, −6.01996448963381507467684273028, −5.17674047526950224944936061318, −4.24087738904113026042717757259, −3.59742406498305828308154772867, −2.32526813465857718722784307905, −1.43734843856762101298947808506,
1.43734843856762101298947808506, 2.32526813465857718722784307905, 3.59742406498305828308154772867, 4.24087738904113026042717757259, 5.17674047526950224944936061318, 6.01996448963381507467684273028, 7.05704044075678616417823809938, 7.67832782283085308061068242045, 8.677792325242591825827135877996, 9.076343222689528847062851881906
