| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 6.24·11-s + 12-s + 5.65·13-s + 16-s − 5.41·17-s + 18-s − 1.17·19-s − 6.24·22-s − 8.82·23-s + 24-s + 5.65·26-s + 27-s + 5.41·29-s − 1.75·31-s + 32-s − 6.24·33-s − 5.41·34-s + 36-s − 8.24·37-s − 1.17·38-s + 5.65·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.88·11-s + 0.288·12-s + 1.56·13-s + 0.250·16-s − 1.31·17-s + 0.235·18-s − 0.268·19-s − 1.33·22-s − 1.84·23-s + 0.204·24-s + 1.10·26-s + 0.192·27-s + 1.00·29-s − 0.315·31-s + 0.176·32-s − 1.08·33-s − 0.928·34-s + 0.166·36-s − 1.35·37-s − 0.190·38-s + 0.905·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.07T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.69448760872775431424330624088, −6.65629407270061499537416067959, −6.22935825226285093271029929896, −5.35064218028725671803783696695, −4.67956150575055488168313288148, −3.89213270331882715943269027458, −3.20665308908475193825144070662, −2.36480568303657116262803294499, −1.68743593710522068391541310608, 0,
1.68743593710522068391541310608, 2.36480568303657116262803294499, 3.20665308908475193825144070662, 3.89213270331882715943269027458, 4.67956150575055488168313288148, 5.35064218028725671803783696695, 6.22935825226285093271029929896, 6.65629407270061499537416067959, 7.69448760872775431424330624088
