L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 5·11-s + 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 21-s − 5·22-s + 6·23-s + 24-s − 26-s + 27-s + 28-s − 10·29-s − 8·31-s + 32-s − 5·33-s − 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.218·21-s − 1.06·22-s + 1.25·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.870·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 59 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.37176374767646101769071395541, −6.90712438389920852567990318866, −5.82997687570592499862067387279, −5.11891702080251256204510452689, −4.84038032324241579421453806352, −3.67308474714850153463830087441, −3.20463113877618829329695428909, −2.27116473412086203430563285280, −1.66449989833715336263004141787, 0,
1.66449989833715336263004141787, 2.27116473412086203430563285280, 3.20463113877618829329695428909, 3.67308474714850153463830087441, 4.84038032324241579421453806352, 5.11891702080251256204510452689, 5.82997687570592499862067387279, 6.90712438389920852567990318866, 7.37176374767646101769071395541