| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 2.58·13-s − 2·15-s + 16-s − 2·17-s + 18-s − 6.24·19-s − 2·20-s − 22-s + 0.828·23-s + 24-s − 25-s − 2.58·26-s + 27-s − 1.65·29-s − 2·30-s + 2.24·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.717·13-s − 0.516·15-s + 0.250·16-s − 0.485·17-s + 0.235·18-s − 1.43·19-s − 0.447·20-s − 0.213·22-s + 0.172·23-s + 0.204·24-s − 0.200·25-s − 0.507·26-s + 0.192·27-s − 0.307·29-s − 0.365·30-s + 0.402·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 0.343T + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 5.41T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 0.485T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−8.215981416779395181291769372963, −7.49882704435399884556820442935, −6.87071793373069035278160071436, −6.03862213101764865665080897013, −4.92910596935009419549359061505, −4.35447120178747222564512163752, −3.60036435677903182752770454037, −2.72038077894336095896560087732, −1.83914839946354303111103362770, 0,
1.83914839946354303111103362770, 2.72038077894336095896560087732, 3.60036435677903182752770454037, 4.35447120178747222564512163752, 4.92910596935009419549359061505, 6.03862213101764865665080897013, 6.87071793373069035278160071436, 7.49882704435399884556820442935, 8.215981416779395181291769372963
