| L(s) = 1 | + 3-s + 3.78·7-s + 9-s − 0.807·11-s + 4.74·13-s − 1.14·17-s + 0.0150·19-s + 3.78·21-s − 6.26·23-s + 27-s + 3.70·29-s − 1.58·31-s − 0.807·33-s − 8.54·37-s + 4.74·39-s + 11.4·41-s + 10.2·43-s − 0.526·47-s + 7.34·49-s − 1.14·51-s − 2.94·53-s + 0.0150·57-s + 11.4·59-s + 3.14·61-s + 3.78·63-s + 13.2·67-s − 6.26·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.43·7-s + 0.333·9-s − 0.243·11-s + 1.31·13-s − 0.278·17-s + 0.00344·19-s + 0.826·21-s − 1.30·23-s + 0.192·27-s + 0.687·29-s − 0.284·31-s − 0.140·33-s − 1.40·37-s + 0.760·39-s + 1.79·41-s + 1.56·43-s − 0.0767·47-s + 1.04·49-s − 0.160·51-s − 0.405·53-s + 0.00198·57-s + 1.48·59-s + 0.402·61-s + 0.477·63-s + 1.62·67-s − 0.754·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.391165030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.391165030\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 + 0.807T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 0.0150T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 0.526T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 + 4.67T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 - 1.42T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 8.06T + 97T^{2} \) |
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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
−7.914249645804773113221881690171, −7.46894278469021518350569614930, −6.49222492546045931547334728476, −5.76824740408138741860483960214, −5.03159925348766166913942250341, −4.17779806398193999941364851208, −3.72134521857237455486263540549, −2.53493385616963170581712281664, −1.85196115175857907189689453849, −0.952713141115918096066760169945,
0.952713141115918096066760169945, 1.85196115175857907189689453849, 2.53493385616963170581712281664, 3.72134521857237455486263540549, 4.17779806398193999941364851208, 5.03159925348766166913942250341, 5.76824740408138741860483960214, 6.49222492546045931547334728476, 7.46894278469021518350569614930, 7.914249645804773113221881690171
